Torkel Franzen argues

[Newberry]

Torkel Franzen argues that all the axioms of ZFC are manifestly true
the logic apparatus is truth preserving therefore all is good and the
system is consistent.

First of all if this is true the the anti-machinists such as Lucas or
Penrose are right because Franzen has just made an argument a machine
cannot do.

So the axioms are manifestly true and the rules are truth preserving.
The argument seems impeccable. What could possibly be wrong with it?
For example is it possible that the logical apparatus contributes some
spurious truths in addition to preserving them? This

(x)((x+3 < x) --> (x = x+4)) (1)

does not look manifestly true to me. Where did it come from? From the
axioms? At this point discussion with the indoctrinated people becomes
difficult. They just repeat that (1) is true under all
interpretations. They are not able to see the problem. In fact this
has nothing to do with any interpretations. The same problem occurs at
the propositional level:

(P & ~P) --> Q (2)

is notoriously counter-intuitive. It is called PARADOX of material
implication, and it motivated research into relevance logics. So don't
tell me that it is all based on manifest truth. In fact I have shown
in another thread
https://groups.google.com/forum/?hl=en&fromgroups#!topic/sci.logic/lDJcgOg4vco
that the proof that the truths of first order arithmetic are not
recursively enumerable is NOT likely to hold if we use Strawson-like
semantics.

[Rupert]

On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>
> the logic apparatus is truth preserving therefore all is good and the
>
> system is consistent.
>
>
>
> First of all if this is true the the anti-machinists such as Lucas or
>
> Penrose are right because Franzen has just made an argument a machine
>
> cannot do.
>

That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.

[Newberry]

That could be done only by another axiomatic system whose consistency
cannot be proven. Franzen's proof is absolute.

A machine cannot perceive that the axioms are manifestly true.

[Nam Nguyen]

What does it mean for a machine to _recognize_ anything, let alone
that a formal system is consistent?

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[Nam Nguyen]

How would you know that a machine can, or can not, _perceive_ something?

[Rupert]

If you were going to encode all the insights available to the AI in a formal system, then yes, Gödel's second incompleteness theorem would apply to that system. That's assuming that you could somehow filter all the errors the AI makes to get a sound formal system. Anyway, so what?

>
>
> A machine cannot perceive that the axioms are manifestly true.

Why not?

[Nam Nguyen]

On 24/04/2013 10:41 PM, Nam Nguyen wrote:
> On 24/04/2013 8:56 PM, Rupert wrote:
>> On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>>>
>>> the logic apparatus is truth preserving therefore all is good and the
>>>
>>> system is consistent.
>>>
>>> First of all if this is true the the anti-machinists such as Lucas or
>>>
>>> Penrose are right because Franzen has just made an argument a machine
>>>
>>> cannot do.
>>>
>> That's nonsense. You've given no good reason at all why we shouldn't
>> one day be able to program an artificial intelligence that can
>> recognize that ZFC is consistent.
>
> What does it mean for a machine to _recognize_ anything, let alone
> that a formal system is consistent?

Iow, does a calculator _recognize_ that 2+2=4?

[Rupert]

On Thursday, April 25, 2013 6:45:35 AM UTC+2, Nam Nguyen wrote:
> On 24/04/2013 9:47 PM, Newberry wrote:
>
> > On Apr 24, 7:56 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >> On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>
> >>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>
> >>
>
> >>> the logic apparatus is truth preserving therefore all is good and the
>
> >>
>
> >>> system is consistent.
>
> >>
>
> >>> First of all if this is true the the anti-machinists such as Lucas or
>
> >>
>
> >>> Penrose are right because Franzen has just made an argument a machine
>
> >>
>
> >>> cannot do.
>
> >>
>
> >> That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
>
> >
>
> > That could be done only by another axiomatic system whose consistency
>
> > cannot be proven. Franzen's proof is absolute.
>
> >
>
> > A machine cannot perceive that the axioms are manifestly true.
>
>
>
> How would you know that a machine can, or can not, _perceive_ something?
>

The only criterion that would be relevant here would be what it tells you.

You could still entertain doubts about whether the machine is "really" conscious, but that's a different issue. We are talking about whether it might be possible to program a machine to be behaviourally indistinguishable from a human mathematician. You've given no good reason why this should not in principle be possible. For example, if it becomes feasible to do a scan of my brain and upload it to a computer, then that would probably do the trick. There doesn't appear to be any reason in principle why such a feat might not become possible in the near future.

[Rupert]

On Thursday, April 25, 2013 6:41:56 AM UTC+2, Nam Nguyen wrote:
> On 24/04/2013 8:56 PM, Rupert wrote:
>
> > On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>
> >> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>
> >>
>
> >> the logic apparatus is truth preserving therefore all is good and the
>
> >>
>
> >> system is consistent.
>
> >>
>
> >>
>
> >>
>
> >> First of all if this is true the the anti-machinists such as Lucas or
>
> >>
>
> >> Penrose are right because Franzen has just made an argument a machine
>
> >>
>
> >> cannot do.
>
> >>
>
> >
>
> > That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
>
>
>
> What does it mean for a machine to _recognize_ anything, let alone
>
> that a formal system is consistent?
>

For my purposes, all I am interested in are the mathematical claims that the machine actually makes. For example, you ask the machine "Is there a natural number that is not a sum of three squares?" and it thinks a while and says "Yes", and then you ask it "Is there a natural number that is not the sum of four squares?", and it thinks a while and comes up with a proof that there is no such natural number. In other words, in its outward manifestations of behaviour it is indistinguishable from a human mathematician. You have given us no reason why this is not in principle possible.

[Nam Nguyen]

On 24/04/2013 10:51 PM, Rupert wrote:
> On Thursday, April 25, 2013 6:45:35 AM UTC+2, Nam Nguyen wrote:
>> On 24/04/2013 9:47 PM, Newberry wrote:
>>
>>> On Apr 24, 7:56 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>>
>>>> On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>>
>>>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>>
>>>>
>>
>>>>> the logic apparatus is truth preserving therefore all is good and the
>>
>>>>
>>
>>>>> system is consistent.
>>
>>>>
>>
>>>>> First of all if this is true the the anti-machinists such as Lucas or
>>
>>>>
>>
>>>>> Penrose are right because Franzen has just made an argument a machine
>>
>>
>>>>> cannot do.
>>
>>>>
>>
>>>> That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
>>
>>
>>> That could be done only by another axiomatic system whose consistency
>>
>>> cannot be proven. Franzen's proof is absolute.
>>
>>>
>>
>>> A machine cannot perceive that the axioms are manifestly true.
>>
>>
>> How would you know that a machine can, or can not, _perceive_ something?
>>
>
> The only criterion that would be relevant here would be what it tells you.

Right. Iow machine _can output_ : stream of bits, signals, what have we.

That's a far cry from _can perceive_ isn't it?

>
> You could still entertain doubts about whether the machine is "really" conscious, but that's a different issue.

It _is_ the issue: whether or not a machine can _perceive_ .

> We are talking about whether it might be possible to program a machine to be behaviourally indistinguishable from a human mathematician.

So you're talking about the output of a machine being homomorphic (if
isomorphic) to a human's output. _Not_ about whether or not a machine
can _perceive_ .

> You've given no good reason why this should not in principle be possible.

But, I've never denied such an isomorphism, homomorphism of
input/output.

> For example, if it becomes feasible to do a scan of my brain and upload it to a computer, then that would probably do the trick. There doesn't appear to be any reason in principle why such a feat might not become possible in the near future.

Again, I'm not denying such an isomorphism, homomorphism.

The point I'm trying to say is that people who argue whether or not
a machine can _perceive_ either aren't carefully in choosing technical
words, or actually don't know what they're talking about.

[Nam Nguyen]

I didn't say anything about _that_ being impossible. I just noticed
that so far you and the other poster have not clearly, formally,
meaningfully, soundly, defined what it'd mean for a computer to have
_perception_ the way a human does.

Not that there isn't: there is a good definition!

[Bill Taylor]

On Apr 25, 1:28 pm, Newberry <newberr...@gmail.com> wrote:

>     (x)((x+3 < x) --> (x = x+4))          (1)
>

> does not look manifestly true to me. ... The same problem occurs

> at the propositional level:
>
>     (P & ~P) --> Q                               (2)
>
> is notoriously counter-intuitive. It is called PARADOX of material
> implication, and it motivated research into relevance logics.

They are only paradoxes for relevant implication.

They are not paradoxes of Boolean implication.

-- Battling Bill

** Creation science - one of the flat earth sciences?

[Rupert]

On Thursday, April 25, 2013 7:08:01 AM UTC+2, Nam Nguyen wrote:
> On 24/04/2013 10:51 PM, Rupert wrote:
>
> > On Thursday, April 25, 2013 6:45:35 AM UTC+2, Nam Nguyen wrote:
>
> >> On 24/04/2013 9:47 PM, Newberry wrote:
>
> >>
>
> >>> On Apr 24, 7:56 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >>
>
> >>>> On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>
> >>
>
> >>>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>
> >>
>
> >>>>
>
> >>
>
> >>>>> the logic apparatus is truth preserving therefore all is good and the
>
> >>
>
> >>>>
>
> >>
>
> >>>>> system is consistent.
>
> >>
>
> >>>>
>
> >>
>
> >>>>> First of all if this is true the the anti-machinists such as Lucas or
>
> >>
>
> >>>>
>
> >>
>
> >>>>> Penrose are right because Franzen has just made an argument a machine
>
> >>
>
> >>
>
> >>>>> cannot do.
>
> >>
>
> >>>>
>
> >>
>
> >>>> That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
>
> >>
>
> >>
>
> >>> That could be done only by another axiomatic system whose consistency
>
> >>
>
> >>> cannot be proven. Franzen's proof is absolute.
>
> >>
>
> >>>
>
> >>
>
> >>> A machine cannot perceive that the axioms are manifestly true.
>
> >>
>
> >>
>
> >> How would you know that a machine can, or can not, _perceive_ something?
>
> >>
>
> >
>
> > The only criterion that would be relevant here would be what it tells you.
>
>
>
> Right. Iow machine _can output_ : stream of bits, signals, what have we.
>
>
>
> That's a far cry from _can perceive_ isn't it?
>

That depends what you mean by "perceive". It's a different issue to the thesis put forward by Lucas and Penrose. They claim that a computer wouldn't even be able to convincingly simulate the behaviour of a human mathematician.

> >
>
> > You could still entertain doubts about whether the machine is "really" conscious, but that's a different issue.
>
>
>
> It _is_ the issue: whether or not a machine can _perceive_ .
>

That is a different issue to the Lucas-Penrose thesis. Their thesis about whether or not a machine can adequately simulate the behaviour of a human mathematician. For the purpose of assessing that thesis, we do not need to worry about whether or not the machine "really" would be conscious.

>
>
> > We are talking about whether it might be possible to program a machine to be behaviourally indistinguishable from a human mathematician.
>
>
>
> So you're talking about the output of a machine being homomorphic (if
>
> isomorphic) to a human's output. _Not_ about whether or not a machine
>
> can _perceive_ .
>

I wasn't talking about that, no. I was talking about the Lucas-Penrose thesis.

>
>
> > You've given no good reason why this should not in principle be possible.
>
>
>
> But, I've never denied such an isomorphism, homomorphism of
>
> input/output.
>

When you endorsed the Lucas-Penrose thesis, you did deny this. If you want to retract that that's fine.

>
>
> > For example, if it becomes feasible to do a scan of my brain and upload it to a computer, then that would probably do the trick. There doesn't appear to be any reason in principle why such a feat might not become possible in the near future.
>
>
>
> Again, I'm not denying such an isomorphism, homomorphism.
>

You did when you endorsed the Lucas-Penrose thesis, but perhaps you were just confused about what the thesis said.

>
>
> The point I'm trying to say is that people who argue whether or not
>
> a machine can _perceive_ either aren't carefully in choosing technical
>
> words, or actually don't know what they're talking about.
>

Well, actually, you've given no evidence for that claim.

[Rupert]

Sorry, I think I confused you with the other poster. Anyway, the point is, I am discussing the Lucas-Penrose thesis, which the other poster endorsed. For the purpose of addressing that thesis it is not necessary to address the issue of whether or not the machine actually would be conscious.

[Rupert]

Well, that's not what we were talking about; we were talking about the Lucas-Penrose thesis.

[Newberry]

On Apr 25, 12:56 am, Rupert <rupertmccal...@yahoo.com> wrote:
> Sorry, I think I confused you with the other poster. Anyway, the point is, I am discussing the Lucas-Penrose thesis, which the other poster endorsed. For the purpose of addressing that thesis it is not necessary to address the issue of whether or not the machine actually would be conscious.

Who endorsed it?

[Newberry]

IF WHAT FRANZEN SAYS IS TRUE then a machine can never prove that ZFC
is consistent. You can prove in T' that T is consistent, i.e. you have
a poff T is consistent provided T' is consistent. Now T' is consistent
provided T" is consistent etc. ad infinitum. You can neverprove the
entire chain. But Franzen has just proved that ZFC is consistent.

How would you program a machine to perceive that the axioms of ZFC are
manifestly true? If you did a contradiction would result.

BTW, my main thesis was that what Franzen says is not true.

[Alan Smaill]

Newberry <newbe...@gmail.com> writes:

> Torkel Franzen argues that all the axioms of ZFC are manifestly true
> the logic apparatus is truth preserving therefore all is good and the
> system is consistent.

Really??

Where did he make this claim?


--
Alan Smaill

[Rupert]

You did, didn't you?

[Rupert]

On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
> On Apr 24, 9:48 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>
> > On Thursday, April 25, 2013 5:47:53 AM UTC+2, Newberry wrote:
>
> > > On Apr 24, 7:56 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >
>
> > > > On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>
> >
>
> > > > > Torkel Franzen argues that all the axioms of ZFC  are manifestly true
>
> >
>
> > > > > the logic apparatus is truth preserving therefore all is good and the
>
> >
>
> > > > > system is consistent.
>
> >
>
> > > > > First of all if this is true the the anti-machinists such as Lucas or
>
> >
>
> > > > > Penrose are right because Franzen has just made an argument a machine
>
> >
>
> > > > > cannot do.
>
> >
>
> > > > That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
>
> >
>
> > > That could be done only by another axiomatic system whose consistency
>
> >
>
> > > cannot be proven. Franzen's proof is absolute.
>
> >
>
> > If you were going to encode all the insights available to the AI in a formal system, then yes, Gödel's second incompleteness theorem would apply to that system. That's assuming that you could somehow filter all the errors the AI makes to get a sound formal system. Anyway, so what?
>
> >
>
> >
>
> >
>
> > > A machine cannot perceive that the axioms are manifestly true.
>
> >
>
> > Why not?
>
>
>
> IF WHAT FRANZEN SAYS IS TRUE then a machine can never prove that ZFC
>
> is consistent.

Why? What do you mean by "prove"?

> You can prove in T' that T is consistent, i.e. you have
>
> a poff T is consistent provided T' is consistent.

You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.

> Now T' is consistent
>
> provided T" is consistent etc. ad infinitum. You can neverprove the
>
> entire chain.

Well, obviously you have to assume some axioms if you want to prove anything at all.

> But Franzen has just proved that ZFC is consistent.
>

When did he do that?

>
>
> How would you program a machine to perceive that the axioms of ZFC are
>
> manifestly true?

What do you mean by that?

> If you did a contradiction would result.
>

Why?

>
>
> BTW, my main thesis was that what Franzen says is not true.

Which thing that he says is not true?

[FredJeffries]

On Apr 25, 8:25 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:

> Newberry <newberr...@gmail.com> writes:
> > Torkel Franzen argues that all the axioms of ZFC  are manifestly true
> > the logic apparatus is truth preserving therefore all is good and the
> > system is consistent.
>
> Really??
>
> Where did he make this claim?

In "The Popular Impact of Gödel's Incompleteness Theorem"

http://www.ams.org/notices/200604/fea-franzen.pdf

he says:

"we can easily, indeed trivially, prove PA consistent using
reasoning of a kind that mathematicians otherwise
use without qualms in proving theorems of
arithmetic. Basically, this easy consistency proof observes
that all theorems of PA are derived by valid
logical reasoning from basic principles true of the
natural numbers, so no contradiction is derivable in PA"

[Ross A. Finlayson]

Neither infinity as axiomatized in ZF nor regularity are obviously
manifestly true. Those restrictions of comprehension where the other
axioms simply expand comprehension don't necessarily reflect, for
example, any anti-foundational sets which some would have as obviously
existant.

Finite combinatorics and Presburger arithmetic are complete (where
unbounded and not necessarily infinite), regular axiomatization, or
rather, axiomatization as regular, of infinity, is disputable.

Regards,

Ross Finlayson

[Newberry]

On Thursday, April 25, 2013 9:27:31 AM UTC-7, Rupert wrote:
> On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
>
> > On Apr 24, 9:48 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >
>
> > > On Thursday, April 25, 2013 5:47:53 AM UTC+2, Newberry wrote:
>
> >
>
> > > > On Apr 24, 7:56 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>
> >
>
> > >
>
> >
>
> > > > > On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>
> >
>
> > >
>
> >
>
> > > > > > Torkel Franzen argues that all the axioms of ZFC  are manifestly true
>
> >
>
> > >
>
> >
>
> > > > > > the logic apparatus is truth preserving therefore all is good and the
>
> >
>
> > >
>
> >
>
> > > > > > system is consistent.
>
> >
>
> > >
>
> >
>
> > > > > > First of all if this is true the the anti-machinists such as Lucas or
>
> >
>
> > >
>
> >
>
> > > > > > Penrose are right because Franzen has just made an argument a machine
>
> >
>
> > >
>
> >
>
> > > > > > cannot do.
>
> >
>
> > >
>
> >
>
> > > > > That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
>
> >
>
> > >
>
> >
>
> > > > That could be done only by another axiomatic system whose consistency
>
> >
>
> > >
>
> >
>
> > > > cannot be proven. Franzen's proof is absolute.
>
> >
>
> > >
>
> >
>
> > > If you were going to encode all the insights available to the AI in a formal system, then yes, Gödel's second incompleteness theorem would apply to that system. That's assuming that you could somehow filter all the errors the AI makes to get a sound formal system. Anyway, so what?
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > A machine cannot perceive that the axioms are manifestly true.
>
> >
>
> > >
>
> >
>
> > > Why not?
>
> >
>
> >
>
> >
>
> > IF WHAT FRANZEN SAYS IS TRUE then a machine can never prove that ZFC
>
> >
>
> > is consistent.
>
>
>
> Why? What do you mean by "prove"?
>
>
>
> > You can prove in T' that T is consistent, i.e. you have
>
> >
>
> > a poff T is consistent provided T' is consistent.
>
>
>
> You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.

So you know what a "proof" is after all. How can a computer believe in T'?

>
>
>
> > Now T' is consistent
>
> >
>
> > provided T" is consistent etc. ad infinitum. You can neverprove the
>
> >
>
> > entire chain.
>
>
>
> Well, obviously you have to assume some axioms if you want to prove anything at all.

Do the axioms have to be true, are we talking about a meaningless game with symbols, or what are you saying here?

>
>
>
> > But Franzen has just proved that ZFC is consistent.
>
> >
>
>
>
> When did he do that?
>
>
>
> >
>
> >
>
> > How would you program a machine to perceive that the axioms of ZFC are
>
> >
>
> > manifestly true?
>
>
>
> What do you mean by that?

Which word you did not understand?

>
>
>
> > If you did a contradiction would result.
>
> >
>
>
>
> Why?

By Gödel's second theorem.

>
>
>
> >
>
> >
>
> > BTW, my main thesis was that what Franzen says is not true.
>
>
>
> Which thing that he says is not true?

My thesis is that the following argument by Torkel Franzen is incorrect:
The axioms of ZFC are manifestly true. The rules of logic are truth preserving. Therefore everything we derive by those rules from those axioms is true. If everything is true then there is no contradiction. Hence ZFC is consistent.
[BTW, I am NOT claiming that ZFC is not consistent.]

[Nam Nguyen]

Your position here is very much the same as that of a Relativist.

Welcome on board! :-)

[Nam Nguyen]

Well, I was talking about that too!

[Rupert]

On Friday, April 26, 2013 3:39:29 AM UTC+2, Newberry wrote:
> On Thursday, April 25, 2013 9:27:31 AM UTC-7, Rupert wrote:
>
> > On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
>
>
> > You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.
>
>
>
> So you know what a "proof" is after all. How can a computer believe in T'?
>

If the computer has cognitive states, then it can believe in T'. You may think a computer can't have cognitive states, but that's not relevant to assessing the Lucas-Penrose thesis. The Lucas-Penrose thesis is just about whether in principle a machine could be programmed to be behaviourally indistinguishable from a human mathematician.

> > Well, obviously you have to assume some axioms if you want to prove anything at all.
>
>
>
> Do the axioms have to be true, are we talking about a meaningless game with symbols, or what are you saying here?
>

You'd be hoping that they were true.

> > > How would you program a machine to perceive that the axioms of ZFC are
>
> >
>
> > >
>
> >
>
> > > manifestly true?
>
> >
>
> >
>
> >
>
> > What do you mean by that?
>
> Which word you did not understand?
>

"Perceive".

> > > If you did a contradiction would result.
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > Why?
>
> By Gödel's second theorem.
>

All that Gödel's second theorem tells you is that if ZFC is consistent, then there is no proof in ZFC that ZFC is consistent. But there is no good reason to suppose that the things that the machine can "perceive" are limited to what can be proved in ZFC.

If there were some formal system F which encoded the insights available to the machine and F were a consistent extension of ZFC, then the machine would be unable to perceive the consistency of F.

> >
>
> >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > BTW, my main thesis was that what Franzen says is not true.
>
> >
>
> >
>
> >
>
> > Which thing that he says is not true?
>
> My thesis is that the following argument by Torkel Franzen is incorrect:
>
> The axioms of ZFC are manifestly true. The rules of logic are truth preserving. Therefore everything we derive by those rules from those axioms is true. If everything is true then there is no contradiction. Hence ZFC is consistent.
>
> [BTW, I am NOT claiming that ZFC is not consistent.]

Do you mean that the argument is not valid, or that one of its premises is false? Which one?

[Rupert]

No, you made it clear that you were not talking about the Lucas-Penrose thesis. You said that you were not saying anything about whether it was possible to do a computer simulation of a human mathematician. That is what the Lucas-Penrose thesis is about. It has nothing to do with what it would mean for a computer to "perceive" things.

[Rupert]

What Franzen is doing there is making the unassailable observation that there is a proof of the consistency of PA in PA extended by a truth predicate. That does not mean that he would have made the philosophical claim that "the axioms of ZFC are manifestly true". This citation provides no evidence that he held that position.

[Nam Nguyen]

On 25/04/2013 11:42 PM, Rupert wrote:
> On Friday, April 26, 2013 4:55:18 AM UTC+2, Nam Nguyen wrote:
>> On 25/04/2013 1:57 AM, Rupert wrote:
>>
>>
>>> Well, that's not what we were talking about; we were talking about the Lucas-Penrose thesis.
>>
>> Well, I was talking about that too!
>>
>
> No, you made it clear that you were not talking about the Lucas-Penrose thesis.

No. You just didn't read the conversation carefully.

> You said that you were not saying anything about whether it was possible to do a computer simulation of a human mathematician.

Where exactly did I say that?

> That is what the Lucas-Penrose thesis is about. It has nothing to do with what it would mean for a computer to "perceive" things.
>

You're wrong. "Human" (your word), "Mind", "Cognition", "Consciousness",
"Perception" is what Lucas-Penrose thesis is _about_ !

[Nam Nguyen]

On 25/04/2013 11:45 PM, Rupert wrote:
> On Thursday, April 25, 2013 9:58:56 PM UTC+2, FredJeffries wrote:
>> On Apr 25, 8:25 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>>
>>> Newberry <newberr...@gmail.com> writes:
>>
>>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>>
>>>> the logic apparatus is truth preserving therefore all is good and the
>>
>>>> system is consistent.
>>
>>
>>> Really??
>>
>>
>>> Where did he make this claim?
>>
>>
>> In "The Popular Impact of Gödel's Incompleteness Theorem"
>>
>>
>> http://www.ams.org/notices/200604/fea-franzen.pdf
>>
>> he says:
>>
>> "we can easily, indeed trivially, prove PA consistent using
>>
>> reasoning of a kind that mathematicians otherwise
>>
>> use without qualms in proving theorems of
>>
>> arithmetic. Basically, this easy consistency proof observes
>>
>> that all theorems of PA are derived by valid
>>
>> logical reasoning from basic principles true of the
>>
>> natural numbers, so no contradiction is derivable in PA"
>
> What Franzen is doing there is making the unassailable observation that there is a proof of the consistency of PA in PA extended by a truth predicate.

You're wrong: what Franzen is doing there is assailable.

[Alan Smaill]

Exactly right, of course.

--
Alan Smaill

[Rupert]

On Friday, April 26, 2013 8:18:05 AM UTC+2, Nam Nguyen wrote:
> On 25/04/2013 11:42 PM, Rupert wrote:
>
> > On Friday, April 26, 2013 4:55:18 AM UTC+2, Nam Nguyen wrote:
>
> >> On 25/04/2013 1:57 AM, Rupert wrote:
>
> >>
>
> >>
>
> >>> Well, that's not what we were talking about; we were talking about the Lucas-Penrose thesis.
>
> >>
>
> >> Well, I was talking about that too!
>
> >>
>
> >
>
> > No, you made it clear that you were not talking about the Lucas-Penrose thesis.
>
>
>
> No. You just didn't read the conversation carefully.
>

Yes, I did.

>
>
> > You said that you were not saying anything about whether it was possible to do a computer simulation of a human mathematician.
>
>
>
> Where exactly did I say that?
>

I can't really be bothered finding the exact citation; I have no doubt you could find it yourself if you just re-read the posts you've made in this thread.

>
>
> > That is what the Lucas-Penrose thesis is about. It has nothing to do with what it would mean for a computer to "perceive" things.
>
> >
>
> You're wrong. "Human" (your word), "Mind", "Cognition", "Consciousness",
>
> "Perception" is what Lucas-Penrose thesis is _about_ !
>

Actually, the Lucas-Penrose thesis is not about consciousness. It is the claim that it would be in principle impossible for a computer to adequately simulate a human mathematician. Questions about whether the simulation would be conscious if it could actually be achieved are a different issue.

[Rupert]

On Friday, April 26, 2013 8:23:45 AM UTC+2, Nam Nguyen wrote:
> On 25/04/2013 11:45 PM, Rupert wrote:
>
> > On Thursday, April 25, 2013 9:58:56 PM UTC+2, FredJeffries wrote:
>
> >> On Apr 25, 8:25 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>
> >>
>
> >>> Newberry <newberr...@gmail.com> writes:
>
> >>
>
> >>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>
> >>
>
> >>>> the logic apparatus is truth preserving therefore all is good and the
>
> >>
>
> >>>> system is consistent.
>
> >>
>
> >>
>
> >>> Really??
>
> >>
>
> >>
>
> >>> Where did he make this claim?
>
> >>
>
> >>
>
> >> In "The Popular Impact of Gödel's Incompleteness Theorem"
>
> >>
>
> >>
>
> >> http://www.ams.org/notices/200604/fea-franzen.pdf
>
> >>
>
> >> he says:
>
> >>
>
> >> "we can easily, indeed trivially, prove PA consistent using
>
> >>
>
> >> reasoning of a kind that mathematicians otherwise
>
> >>
>
> >> use without qualms in proving theorems of
>
> >>
>
> >> arithmetic. Basically, this easy consistency proof observes
>
> >>
>
> >> that all theorems of PA are derived by valid
>
> >>
>
> >> logical reasoning from basic principles true of the
>
> >>
>
> >> natural numbers, so no contradiction is derivable in PA"
>
> >
>
> > What Franzen is doing there is making the unassailable observation that there is a proof of the consistency of PA in PA extended by a truth predicate.
>
>
>
> You're wrong: what Franzen is doing there is assailable.
>

No, it's not. The observation is that if you extend the language of PA to include a truth predicate and add certain axioms for that truth predicate together with induction axioms for formulas involving the truth predicate, then in the resulting system the consistency of PA is provable. This is a mathematical theorem which is not subject to reasonable doubt.

[Peter Percival]

Nam Nguyen wrote:
>
> On 24/04/2013 9:47 PM, Newberry wrote:

> > On Apr 24, 7:56 pm, Rupert <rupertmccal...@yahoo.com> wrote:
> >> On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:

> >>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
> >>
> >>> the logic apparatus is truth preserving therefore all is good and the
> >>
> >>> system is consistent.
> >>

> >>> First of all if this is true the the anti-machinists such as Lucas or
> >>
> >>> Penrose are right because Franzen has just made an argument a machine
> >>
> >>> cannot do.
> >>
> >> That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
> >

> > That could be done only by another axiomatic system whose consistency
> > cannot be proven. Franzen's proof is absolute.
> >

> > A machine cannot perceive that the axioms are manifestly true.
>

> How would you know that a machine can, or can not, _perceive_ something?

Humans are machines, and they can perceive.

--
Thou hast committed--
Fornication? But that was in another country:
and besides, the wench is dead.
(Marlow, The Jew of Malta, act 4, scene 1.)

[Peter Percival]

Nam Nguyen wrote:

> The point I'm trying to say is that people who argue whether or not
> a machine can _perceive_ either aren't carefully in choosing technical
> words, or actually don't know what they're talking about.

People often don't know what they are talking about. It is uncharitable
to hold it against them.

[Nam Nguyen]

On 26/04/2013 4:04 AM, Rupert wrote:
> On Friday, April 26, 2013 8:18:05 AM UTC+2, Nam Nguyen wrote:
>> On 25/04/2013 11:42 PM, Rupert wrote:

>>> No, you made it clear that you were not talking about the Lucas-Penrose thesis.
>>
>> No. You just didn't read the conversation carefully.
>>
>
> Yes, I did.

No, you didn't.

>
>>
>>> You said that you were not saying anything about whether it was possible to do a computer simulation of a human mathematician.
>>
>> Where exactly did I say that?
>>
>
> I can't really be bothered finding the exact citation; I have no doubt you could find it yourself if you just re-read the posts you've made in this thread.

I did, and it showed you had not read the conversation carefully!

>
>>
>>
>>> That is what the Lucas-Penrose thesis is about. It has nothing to do with what it would mean for a computer to "perceive" things.
>>
>>
>> You're wrong. "Human" (your word), "Mind", "Cognition", "Consciousness",
>>
>> "Perception" is what Lucas-Penrose thesis is _about_ !
>>
>
> Actually, the Lucas-Penrose thesis is not about consciousness. It is the claim that it would be in principle impossible for a computer to adequately simulate a human mathematician.

I see. So, Lucas-Penrose thesis is specifically about "a computer to
adequately simulate a human" but it's not about "Human" (your word),
"Mind", "Cognition", "Consciousness", "Perception", which are all
important attributes of human computation ability?

It's evident that you don't know what you're defending, talking about.

> Questions about whether the simulation would be conscious if it could actually be achieved are a different issue.
>

How would you define "adequately simulate a human mathematician",
so that you'd have a _different_ simulation issue?

[Nam Nguyen]

On 26/04/2013 4:55 AM, Peter Percival wrote:
> Nam Nguyen wrote:
>>
>> On 24/04/2013 9:47 PM, Newberry wrote:
>>> On Apr 24, 7:56 pm, Rupert <rupertmccal...@yahoo.com> wrote:
>>>> On Thursday, April 25, 2013 3:28:58 AM UTC+2, Newberry wrote:
>>>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>>>>
>>>>> the logic apparatus is truth preserving therefore all is good and the
>>>>
>>>>> system is consistent.
>>>>
>>>>> First of all if this is true the the anti-machinists such as Lucas or
>>>>
>>>>> Penrose are right because Franzen has just made an argument a machine
>>>>
>>>>> cannot do.
>>>>
>>>> That's nonsense. You've given no good reason at all why we shouldn't one day be able to program an artificial intelligence that can recognize that ZFC is consistent.
>>>
>>> That could be done only by another axiomatic system whose consistency
>>> cannot be proven. Franzen's proof is absolute.
>>>
>>> A machine cannot perceive that the axioms are manifestly true.
>>
>> How would you know that a machine can, or can not, _perceive_ something?
>
> Humans are machines, and they can perceive.

What about the other _non-human_ machines, such as a computer, a
calculator?

So, the calculator would perceive the arithmetic truth of say 2+2=4,
_as a human being_ ?

[Nam Nguyen]

Care to define what a "truth predicate" be (in your defending what
Franzen said be unassailable) ?

[Rupert]

On Friday, April 26, 2013 5:41:44 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 4:04 AM, Rupert wrote:
>
> > On Friday, April 26, 2013 8:18:05 AM UTC+2, Nam Nguyen wrote:
>
> >> On 25/04/2013 11:42 PM, Rupert wrote:
>
>
>
> >>> No, you made it clear that you were not talking about the Lucas-Penrose thesis.
>
> >>
>
> >> No. You just didn't read the conversation carefully.
>
> >>
>
> >
>
> > Yes, I did.
>
>
>
> No, you didn't.
>

Your opinion lacks rational foundation.

> >
>
> >>
>
> >>> You said that you were not saying anything about whether it was possible to do a computer simulation of a human mathematician.
>
> >>
>
> >> Where exactly did I say that?
>
> >>
>
> >
>
> > I can't really be bothered finding the exact citation; I have no doubt you could find it yourself if you just re-read the posts you've made in this thread.
>
>
>
> I did, and it showed you had not read the conversation carefully!
>

In this post

https://groups.google.com/group/sci.logic/msg/6414ca25e0c89b61?dmode=source&output=gplain&noredirect&pli=1

you responded to my writing

"In other words, in its outward manifestations of behaviour it is indistinguishable from a human mathematician. You have given us no reason why this is not in principle possible."

by saying

"I didn't say anything about _that_ being impossible."

As we can see, I was absolutely correct when I said that you indicated that you had not said anything about the possibility of doing a computer simulation of a human mathematician.

> >
>
> >>
>
> >>
>
> >>> That is what the Lucas-Penrose thesis is about. It has nothing to do with what it would mean for a computer to "perceive" things.
>
> >>
>
> >>
>
> >> You're wrong. "Human" (your word), "Mind", "Cognition", "Consciousness",
>
> >>
>
> >> "Perception" is what Lucas-Penrose thesis is _about_ !
>
> >>
>
> >
>
> > Actually, the Lucas-Penrose thesis is not about consciousness. It is the claim that it would be in principle impossible for a computer to adequately simulate a human mathematician.
>
>
>
> I see. So, Lucas-Penrose thesis is specifically about "a computer to
>
> adequately simulate a human" but it's not about "Human" (your word),
>
> "Mind", "Cognition", "Consciousness", "Perception", which are all
>
> important attributes of human computation ability?
>

It's not about consciousness.

>
>
> It's evident that you don't know what you're defending, talking about.
>

You think?

>
>
> > Questions about whether the simulation would be conscious if it could actually be achieved are a different issue.
>
> >
>
>
>
> How would you define "adequately simulate a human mathematician",
>
> so that you'd have a _different_ simulation issue?
>

Which word don't you understand?

[Rupert]

On Friday, April 26, 2013 5:57:03 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 4:05 AM, Rupert wrote:
>
> > On Friday, April 26, 2013 8:23:45 AM UTC+2, Nam Nguyen wrote:
>
> >> On 25/04/2013 11:45 PM, Rupert wrote:
>
> >>
>
> >>> What Franzen is doing there is making the unassailable observation that there is a proof of the consistency of PA in PA extended by a truth predicate.
>
> >>
>
> >> You're wrong: what Franzen is doing there is assailable.
>
> >>
>
> >
>
> > No, it's not. The observation is that if you extend the language of PA to include a truth predicate and add certain axioms for that truth predicate together with induction axioms for formulas involving the truth predicate, then in the resulting system the consistency of PA is provable. This is a mathematical theorem which is not subject to reasonable doubt.
>
>
>
> Care to define what a "truth predicate" be (in your defending what
>
> Franzen said be unassailable) ?
>

It is a unary predicate which you add to the first-order language of arithmetic. The intended interpretation is that this predicate holds of a natural number if and only if that natural number is the Gödel number of a sentence in the first-order language of arithmetic which is true in the standard model. Then you add to PA some axioms about the truth predicate which basically amount to spelling out the Tarski clauses, and also all the induction axioms for formulas involving the truth predicate. In the resulting theory, the consistency of PA is provable.

Is that sufficiently clear?

[Nam Nguyen]

On 26/04/2013 4:57 AM, Peter Percival wrote:
> Nam Nguyen wrote:
>
>> The point I'm trying to say is that people who argue whether or not
>> a machine can _perceive_ either aren't carefully in choosing technical
>> words, or actually don't know what they're talking about.
>
> People often don't know what they are talking about. It is uncharitable
> to hold it against them.

Not that anyone is desiring uncharitable aspect of a mathematical
_logic_ argument, but we live in a binary logic world where assertions
must necessarily be true or false, correct or incorrect.

Especially if one cares to claim someone else as being wrong or
incorrect, then one must necessarily know what one is talking about.

There's nothing around that, except following Wittgenstein's advise:

"Wovon man nicht sprechen kann, dar�ber muss man schweigen"

[Nam Nguyen]

On 26/04/2013 9:58 AM, Rupert wrote:
> On Friday, April 26, 2013 5:41:44 PM UTC+2, Nam Nguyen wrote:
>> On 26/04/2013 4:04 AM, Rupert wrote:
>>
>>> On Friday, April 26, 2013 8:18:05 AM UTC+2, Nam Nguyen wrote:
>>
>>>> On 25/04/2013 11:42 PM, Rupert wrote:
>>
>>>>> No, you made it clear that you were not talking about the Lucas-Penrose thesis.
>>
>>
>>>> No. You just didn't read the conversation carefully.
>>
>>> Yes, I did.
>>
>>
>> No, you didn't.
>>
>
> Your opinion lacks rational foundation.
>
>>
>>>>> You said that you were not saying anything about whether it was possible to do a computer simulation of a human mathematician.
>>
>>
>>>> Where exactly did I say that?
>>
>>> I can't really be bothered finding the exact citation; I have no doubt you could find it yourself if you just re-read the posts you've made in this thread.
>>
>>
>> I did, and it showed you had not read the conversation carefully!
>>
>
> In this post
>
> https://groups.google.com/group/sci.logic/msg/6414ca25e0c89b61?dmode=source&output=gplain&noredirect&pli=1
>
> you responded to my writing
>
> "In other words, in its outward manifestations of behaviour it is indistinguishable from a human mathematician. You have given us no reason why this is not in principle possible."
>
> by saying
>
> "I didn't say anything about _that_ being impossible."

See. That points to your mistake: you didn't quote enough from that post
to show your knee-jerk reaction reading. See below.

>
> As we can see, I was absolutely correct when I said that you indicated that you had not said anything about the possibility of doing a computer simulation of a human mathematician.

As the world can see, you're absolutely wrong.

You said in that post, apparently _to Newberry_ :

>> In other words, in its outward manifestations of behaviour it is
>> indistinguishable from a human mathematician. You have given us no
>> reason why this is not in principle possible.

To which I responded:

> I didn't say anything about _that_ being impossible.

See. I only alerted to you that it was Newberry who had stated something
that looked contradictory to your position (your "You have given
us no reason why this is not in principle possible").

Iow, I was simply making a caveat to you that your "You have given
no reason" shouldn't be referring to me.

I did _NOT_ reveral my position on Lucas-Penrose thesis _there_ (on that
statement you quoted).

> I just noticed that so far you and the other poster have not clearly,
> formally, meaningfully, soundly, defined what it'd mean for a
> computer to have _perception_ the way a human does.

See. Note my "the other poster" (who is Newberry).

Again, I'm absolutly correct that you did _NOT_ read the conversation
carefully.

>>
>>>>> That is what the Lucas-Penrose thesis is about. It has nothing to do with what it would mean for a computer to "perceive" things.
>>
>>>>
>>
>>>>
>>
>>>> You're wrong. "Human" (your word), "Mind", "Cognition", "Consciousness",
>>
>>>>
>>
>>>> "Perception" is what Lucas-Penrose thesis is _about_ !
>>
>>>>
>>
>>>
>>
>>> Actually, the Lucas-Penrose thesis is not about consciousness. It is the claim that it would be in principle impossible for a computer to adequately simulate a human mathematician.
>>
>>
>>
>> I see. So, Lucas-Penrose thesis is specifically about "a computer to
>>
>> adequately simulate a human" but it's not about "Human" (your word),
>>
>> "Mind", "Cognition", "Consciousness", "Perception", which are all
>>
>> important attributes of human computation ability?
>>
>
> It's not about consciousness.
>
>>
>> It's evident that you don't know what you're defending, talking about.
>>
>
> You think?

I don't just think. I know with proof that you don't know what you're
defending, talking about, as I've just proven above.

>>> Questions about whether the simulation would be conscious if it could actually be achieved are a different issue.
>>
>>
>> How would you define "adequately simulate a human mathematician",
>> so that you'd have a _different_ simulation issue?
>>
>
> Which word don't you understand?

The non-existing words of your non-existing _technical definition_
of what "adequately simulate a human mathematician" would _technically_
_entail_ in the context of a mathematical logic discourse.

[Herman Rubin]

On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
> On Apr 25, 8:25�am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>> Newberry <newberr...@gmail.com> writes:
>> > Torkel Franzen argues that all the axioms of ZFC �are manifestly true
>> > the logic apparatus is truth preserving therefore all is good and the
>> > system is consistent.

>> Really??

>> Where did he make this claim?

> In "The Popular Impact of G�del's Incompleteness Theorem"

> http://www.ams.org/notices/200604/fea-franzen.pdf

> he says:

> "we can easily, indeed trivially, prove PA consistent using
> reasoning of a kind that mathematicians otherwise
> use without qualms in proving theorems of
> arithmetic. Basically, this easy consistency proof observes
> that all theorems of PA are derived by valid
> logical reasoning from basic principles true of the
> natural numbers, so no contradiction is derivable in PA"

Mathematicians are willing to assume PA is consistent. The
inconsistency of PA would mean that the basic principles
of the natural numbers are inconsistent.

I recommend that the discussion of the natural numbers from
the basic principles be taught very early, and addition, etc.,
be derived from them. They LOOK obvious. But if it is
consistent, we know we cannot prove it.

Now PA has been proved consistent in ZF or NBG, but then that
brings the consistency of axioms for set theory.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

[Nam Nguyen]

On 26/04/2013 10:00 AM, Rupert wrote:
> On Friday, April 26, 2013 5:57:03 PM UTC+2, Nam Nguyen wrote:
>> On 26/04/2013 4:05 AM, Rupert wrote:
>>
>>> On Friday, April 26, 2013 8:23:45 AM UTC+2, Nam Nguyen wrote:
>>
>>>> On 25/04/2013 11:45 PM, Rupert wrote:
>>
>>
>>>>> What Franzen is doing there is making the unassailable observation that there is a proof of the consistency of PA in PA extended by a truth predicate.
>>
>>>> You're wrong: what Franzen is doing there is assailable.
>>
>>
>>> No, it's not. The observation is that if you extend the language of PA to include a truth predicate and add certain axioms for that truth predicate together with induction axioms for formulas involving the truth predicate, then in the resulting system the consistency of PA is provable. This is a mathematical theorem which is not subject to reasonable doubt.
>>
>> Care to define what a "truth predicate" be (in your defending what
>>
>> Franzen said be unassailable) ?
>>
>

> It is a unary predicate which you add to the first-order language of arithmetic. The intended interpretation is that this predicate holds of a natural number if and only if that natural number is the G�del number of a sentence in the first-order language of arithmetic which is true in the standard model. Then you add to PA some axioms about the truth predicate which basically amount to spelling out the Tarski clauses, and also all the induction axioms for formulas involving the truth predicate. In the resulting theory, the consistency of PA is provable.
>
> Is that sufficiently clear?

Of course not. Everyone could logically see that such an utterance
isn't clear at all, on multiple counts.

(a) You don't know - at minimum up to this moment - and nobody knows
what the "natural numbers", "standard structure [model]" be
when there's a formula (e.g. cGC) that can't be asserted as true or
false (and there are at least some good rationale that it's
logically impossible to know so).

(b) It's nonsensical to assert the consistency of PA outside
the technical definition of formal system consistency which
would involve unprovability _IN_ the underlying system,
PA in this case.

Your alluded "In the resulting theory" isn't PA, hence it's not
just insufficiency clear but also dubious as well as logically
irrelevant as far as FOL _definition_ of formal system consistency.

Of course, if you don't follow FOL _definition_ of formal system
consistency, it will be logically unclear what you were talking
about in your "In the resulting theory, the consistency of PA is provable".

[Nam Nguyen]

On 26/04/2013 10:37 AM, Herman Rubin wrote:
> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
>> On Apr 25, 8:25 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>>> Newberry <newberr...@gmail.com> writes:
>>>> Torkel Franzen argues that all the axioms of ZFC are manifestly true
>>>> the logic apparatus is truth preserving therefore all is good and the
>>>> system is consistent.
>
>>> Really??
>
>>> Where did he make this claim?
>
>
>> In "The Popular Impact of G�del's Incompleteness Theorem"
>
>> http://www.ams.org/notices/200604/fea-franzen.pdf
>
>> he says:
>
>> "we can easily, indeed trivially, prove PA consistent using
>> reasoning of a kind that mathematicians otherwise
>> use without qualms in proving theorems of
>> arithmetic. Basically, this easy consistency proof observes
>> that all theorems of PA are derived by valid
>> logical reasoning from basic principles true of the
>> natural numbers, so no contradiction is derivable in PA"
>
> Mathematicians are willing to assume PA is consistent.

Agree. For the record I've always assumed PA is consistent,
until of course if one day ones present a proof _IN_ PA
of the form (F /\ ~F).

But assumption is _not_ assertion, wouldn't you agree?

(That's all I'm asking.)

> The
> inconsistency of PA would mean that the basic principles
> of the natural numbers are inconsistent.
>
> I recommend that the discussion of the natural numbers from
> the basic principles be taught very early, and addition, etc.,
> be derived from them. They LOOK obvious. But if it is
> consistent, we know we cannot prove it.
>
> Now PA has been proved consistent in ZF or NBG, but then that
> brings the consistency of axioms for set theory.
>


--

[Rupert]

On Friday, April 26, 2013 6:36:06 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 9:58 AM, Rupert wrote:
>
> > On Friday, April 26, 2013 5:41:44 PM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 4:04 AM, Rupert wrote:
>
> > In this post
>
> >
>
> > https://groups.google.com/group/sci.logic/msg/6414ca25e0c89b61?dmode=source&output=gplain&noredirect&pli=1
>
> >
>
> > you responded to my writing
>
> >
>
> > "In other words, in its outward manifestations of behaviour it is indistinguishable from a human mathematician. You have given us no reason why this is not in principle possible."
>
> >
>
> > by saying
>
> >
>
> > "I didn't say anything about _that_ being impossible."
>
>
>
> See. That points to your mistake: you didn't quote enough from that post
>
> to show your knee-jerk reaction reading. See below.
>
> >
>
> > As we can see, I was absolutely correct when I said that you indicated that you had not said anything about the possibility of doing a computer simulation of a human mathematician.
>
>
>
> As the world can see, you're absolutely wrong.
>
>
>
> You said in that post, apparently _to Newberry_ :
>
>
>
> >> In other words, in its outward manifestations of behaviour it is
>
> >> indistinguishable from a human mathematician. You have given us no
>
> >> reason why this is not in principle possible.
>
>
>
> To which I responded:
>
>
>
> > I didn't say anything about _that_ being impossible.
>
>
>
> See. I only alerted to you that it was Newberry who had stated something
>
> that looked contradictory to your position (your "You have given
>
> us no reason why this is not in principle possible").
>

Do you or do you not believe that it is possible for a computer to simulate a human mathematician? Or do you have no opinion about the matter?

>
>
> Iow, I was simply making a caveat to you that your "You have given
>
> no reason" shouldn't be referring to me.
>

Right, I see. I apologize; I was confused between you and Newberry for a while.

>
>
> I did _NOT_ reveral my position on Lucas-Penrose thesis _there_ (on that
>
> statement you quoted).
>

Okay, so I misunderstood.

>
>
> > I just noticed that so far you and the other poster have not clearly,
>
> > formally, meaningfully, soundly, defined what it'd mean for a
>
> > computer to have _perception_ the way a human does.
>
>
>
> See. Note my "the other poster" (who is Newberry).
>

Yes, but my point stands that issues about "perception" have nothing to do with the Lucas-Penrose thesis.

>
>
> Again, I'm absolutly correct that you did _NOT_ read the conversation
>
> carefully.
>

Not really, my misunderstanding was excusable.

>
>
> >>
>
> >>>>> That is what the Lucas-Penrose thesis is about. It has nothing to do with what it would mean for a computer to "perceive" things.
>
> >>
>
> >>>>
>
> >>
>
> >>>>
>
> >>
>
> >>>> You're wrong. "Human" (your word), "Mind", "Cognition", "Consciousness",
>
> >>
>
> >>>>
>
> >>
>
> >>>> "Perception" is what Lucas-Penrose thesis is _about_ !
>
> >>
>
> >>>>
>
> >>
>
> >>>
>
> >>
>
> >>> Actually, the Lucas-Penrose thesis is not about consciousness. It is the claim that it would be in principle impossible for a computer to adequately simulate a human mathematician.
>
> >>
>
> >>
>
> >>
>
> >> I see. So, Lucas-Penrose thesis is specifically about "a computer to
>
> >>
>
> >> adequately simulate a human" but it's not about "Human" (your word),
>
> >>
>
> >> "Mind", "Cognition", "Consciousness", "Perception", which are all
>
> >>
>
> >> important attributes of human computation ability?
>
> >>
>
> >
>
> > It's not about consciousness.
>
> >
>
> >>
>
> >> It's evident that you don't know what you're defending, talking about.
>
> >>
>
> >
>
> > You think?
>
>
>
> I don't just think. I know with proof that you don't know what you're
>
> defending, talking about, as I've just proven above.
>

You didn't do any such thing.

>
>
> >>> Questions about whether the simulation would be conscious if it could actually be achieved are a different issue.
>
> >>
>
> >>
>
> >> How would you define "adequately simulate a human mathematician",
>
> >> so that you'd have a _different_ simulation issue?
>
> >>
>
> >
>
> > Which word don't you understand?
>
>
>
> The non-existing words of your non-existing _technical definition_
>
> of what "adequately simulate a human mathematician" would _technically_
>
> _entail_ in the context of a mathematical logic discourse.
>

Are you familiar with the Turing test?

[Rupert]

On Friday, April 26, 2013 6:59:30 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 10:00 AM, Rupert wrote:
>
> > On Friday, April 26, 2013 5:57:03 PM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 4:05 AM, Rupert wrote:
>
> >>
>
> >>> On Friday, April 26, 2013 8:23:45 AM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>>> On 25/04/2013 11:45 PM, Rupert wrote:
>
> >>
>
> >>
>
> >>>>> What Franzen is doing there is making the unassailable observation that there is a proof of the consistency of PA in PA extended by a truth predicate.
>
> >>
>
> >>>> You're wrong: what Franzen is doing there is assailable.
>
> >>
>
> >>
>
> >>> No, it's not. The observation is that if you extend the language of PA to include a truth predicate and add certain axioms for that truth predicate together with induction axioms for formulas involving the truth predicate, then in the resulting system the consistency of PA is provable. This is a mathematical theorem which is not subject to reasonable doubt.
>
> >>
>
> >> Care to define what a "truth predicate" be (in your defending what
>
> >>
>
> >> Franzen said be unassailable) ?
>
> >>
>
> >
>

> > It is a unary predicate which you add to the first-order language of arithmetic. The intended interpretation is that this predicate holds of a natural number if and only if that natural number is the Gödel number of a sentence in the first-order language of arithmetic which is true in the standard model. Then you add to PA some axioms about the truth predicate which basically amount to spelling out the Tarski clauses, and also all the induction axioms for formulas involving the truth predicate. In the resulting theory, the consistency of PA is provable.

>
> >
>
> > Is that sufficiently clear?
>
>
>
> Of course not. Everyone could logically see that such an utterance
>
> isn't clear at all, on multiple counts.
>
>
>
> (a) You don't know - at minimum up to this moment - and nobody knows
>
> what the "natural numbers", "standard structure [model]" be
>
> when there's a formula (e.g. cGC) that can't be asserted as true or
>
> false (and there are at least some good rationale that it's
>
> logically impossible to know so).
>

There is no good rationale at all for your claim that it is logically impossible to know the truth-value of cGC.

In any event, this is irrelevant. I only made remarks about what the "intended interpretation" of the unary predicate was to help your intuition about what axioms I wanted to postulate for it. I could give a complete definition of the formal theory I want to talk about easily enough without making any reference at all to the notion of "truth in the standard model". My claim that the consistency of PA is a theorem of this formal theory would then be a matter that is in principle machine-checkable.

>
>
> (b) It's nonsensical to assert the consistency of PA outside
>
> the technical definition of formal system consistency which
>
> would involve unprovability _IN_ the underlying system,
>
> PA in this case.
>

I have absolutely no idea what you are babbling on about. There is a certain sentence in the first-order language of arithmetic which we may call the consistency sentence for PA. This sentence is a theorem of the formal theory which I was talking about. As I said, this can all be verified by a feasible computation.

>
>
> Your alluded "In the resulting theory" isn't PA, hence it's not
>
> just insufficiency clear but also dubious as well as logically
>
> irrelevant as far as FOL _definition_ of formal system consistency.
>

I have no idea what you're babbling on about.

My claim is simply that a certain sentence is provable in a certain precisely defined recursively axiomatizable formal theory.

>
>
> Of course, if you don't follow FOL _definition_ of formal system
>
> consistency, it will be logically unclear what you were talking
>
> about in your "In the resulting theory, the consistency of PA is provable".
>

I'm sorry you find it unclear. It's actually a perfectly precise and machine-checkable statement. I'm not sure whether I've got the energy to try to get you to understand.

[Nam Nguyen]

On 26/04/2013 11:18 AM, Rupert wrote:
> On Friday, April 26, 2013 6:59:30 PM UTC+2, Nam Nguyen wrote:

>> (a) You don't know - at minimum up to this moment - and nobody knows
>>
>> what the "natural numbers", "standard structure [model]" be
>>
>> when there's a formula (e.g. cGC) that can't be asserted as true or
>>
>> false (and there are at least some good rationale that it's
>>
>> logically impossible to know so).
>>
>
> There is no good rationale at all for your claim that it is logically impossible to know the truth-value of cGC.

Considering last time we were on the issue, you couldn't defend,
explain, what you would mean by a language structure being an
incoherent concept, three's no rationale to make sense of your
utterance above.

>
> In any event, this is irrelevant.

Of course it's relevant. The natural numbers with all of its "baggage"
about arithmetic truth-relativity would be relevant to the purported
consistency of PA we've been arguing about here and elsewhere.

Didn't you once erroneously state that if PA is inconsistent, the
language structure known "the natural numbers" is incoherent?

> I only made remarks about what the "intended interpretation" of the unary predicate was to help your intuition about what axioms I wanted to postulate for it.

But what's _your definition_ of "intended interpretation" and how would
that help others to understand the "proof" that PA is consistent?

> I could give a complete definition of the formal theory I want to talk about easily enough without making any reference at all to the notion of "truth in the standard model".

> My claim that the consistency of PA is a theorem of this formal theory would then be a matter that is in principle machine-checkable.

It's a trivial observation that a machine can check syntactical
theorems, including those of the form F /\ ~F. How would you go
from there to _claim_ the consistency of PA, something that per
_definition of consistency_ is logically impossible to prove?

>
>>
>>
>> (b) It's nonsensical to assert the consistency of PA outside
>>
>> the technical definition of formal system consistency which
>>
>> would involve unprovability _IN_ the underlying system,
>>
>> PA in this case.
>>
>
> I have absolutely no idea what you are babbling on about. There is a certain sentence in the first-order language of arithmetic which we may call the consistency sentence for PA.

Ah, so that's it! The consistency of PA would technically rest with what
Rupert "may call the consistency sentence for PA"!

And that would be called mathematical _logic_ ?

> This sentence is a theorem of the formal theory which I was talking about. As I said, this can all be verified by a feasible computation.

Sure. Giving me any sentence whatsoever and within seconds I'll come up
with a formal theory that would prove it and of course can be "verified
by a feasible computation" according to you.

That's _easy_ .

>
>>
>>
>> Your alluded "In the resulting theory" isn't PA, hence it's not
>>
>> just insufficiency clear but also dubious as well as logically
>>
>> irrelevant as far as FOL _definition_ of formal system consistency.
>>
>
> I have no idea what you're babbling on about.
>
> My claim is simply that a certain sentence is provable in a certain precisely defined recursively axiomatizable formal theory.

I'm not babbling. I'm just telling you that you're bluffing in saying
that the _real_ consistency of a formal system can be syntactically
be proven by another formal system.

It simply can't be.

>> Of course, if you don't follow FOL _definition_ of formal system
>>
>> consistency, it will be logically unclear what you were talking
>>
>> about in your "In the resulting theory, the consistency of PA is provable".
>>
>
> I'm sorry you find it unclear. It's actually a perfectly precise and machine-checkable statement. I'm not sure whether I've got the energy to try to get you to understand.

Your energy should be saved for your understanding that a "machine
checkable" might guarantee an inconsistency but would have _NO_
say in consistency.

And that's a very simple observation in FOL reasoning that one
should know.

[Nam Nguyen]

Of course I do. My opinion is that Lucas-Penrose thesis is wrong.
And I did explain why.

>>
>> Iow, I was simply making a caveat to you that your "You have given
>> no reason" shouldn't be referring to me.
>>
>
> Right, I see. I apologize; I was confused between you and Newberry for a while.

>> I did _NOT_ reveral my position on Lucas-Penrose thesis _there_ (on that
>>
>> statement you quoted).
>>
> Okay, so I misunderstood.

Then, iirc, your have the same opinion that Lucas-Penrose thesis is
wrong. Right?

>> > I just noticed that so far you and the other poster have not clearly,
>>
>> > formally, meaningfully, soundly, defined what it'd mean for a
>>
>> > computer to have _perception_ the way a human does.

>> See. Note my "the other poster" (who is Newberry).
>>
>
> Yes, but my point stands that issues about "perception" have nothing to do with the Lucas-Penrose thesis.

But _there_ you're wrong. Human intelligence, perception of
mathematical truth, e.g., is the corner stone of the thesis.

Whether or not you attack or defend the thesis, you can not escape this
"hard" issue, as it's often called (iirc).

> Are you familiar with the Turing test?
>

Yes. What about it here?

[Nam Nguyen]

Exactly right. And exactly my point.

Somewhere, somehow, a circularity or an infinite regression
of _mathematical knowledge_ will be reached, and at that point
we still have to confront with the issue of mathematical relativity.

There's really no escape to it, I'm afraid from what I could gather.

[Rupert]

On Friday, April 26, 2013 10:04:18 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 11:14 AM, Rupert wrote:
>
> > On Friday, April 26, 2013 6:36:06 PM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 9:58 AM, Rupert wrote:
>
> >>
>
> >>> On Friday, April 26, 2013 5:41:44 PM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>>> On 26/04/2013 4:04 AM, Rupert wrote:
>
>
> > Do you or do you not believe that it is possible for a computer to simulate a human mathematician? Or do you have no opinion about the matter?
>
>
>
> Of course I do. My opinion is that Lucas-Penrose thesis is wrong.
>
> And I did explain why.
>

Okay, well thank you for clarifying that. You do understand that what you are saying here is that you do think that it is possible for a computer to simulate a human mathematician? That's your view?

>
>
> >>
>
> >> Iow, I was simply making a caveat to you that your "You have given
>
> >> no reason" shouldn't be referring to me.
>
> >>
>
> >
>
> > Right, I see. I apologize; I was confused between you and Newberry for a while.
>
>
>
> >> I did _NOT_ reveral my position on Lucas-Penrose thesis _there_ (on that
>
> >>
>
> >> statement you quoted).
>
> >>
>
> > Okay, so I misunderstood.
>
>
>
> Then, iirc, your have the same opinion that Lucas-Penrose thesis is
>
> wrong. Right?
>

Well, yeah, I mean, I wouldn't claim to know whether or not it is in principle possible for a computer to simulate a human mathematician, but I would think that a whole brain emulation would probably do it, and I think that Lucas and Penrose don't adequtaely defend their view that it's not possible.

>
>
> >> > I just noticed that so far you and the other poster have not clearly,
>
> >>
>
> >> > formally, meaningfully, soundly, defined what it'd mean for a
>
> >>
>
> >> > computer to have _perception_ the way a human does.
>
>
>
> >> See. Note my "the other poster" (who is Newberry).
>
> >>
>
> >
>
> > Yes, but my point stands that issues about "perception" have nothing to do with the Lucas-Penrose thesis.
>
>
>
> But _there_ you're wrong. Human intelligence, perception of
>
> mathematical truth, e.g., is the corner stone of the thesis.
>

Penrose claims that our capacity to "understand" mathematics plays an important role in our ability to do mathematics, and he believes that this is closely tied up with our capacity for conscious awareness. (What Lucas' views were I do not know.) Nevertheless, the thesis itself is just about the question of whether or not a computer could in principle simulate a human mathematician. That in itself has nothing to do with conscious awareness. Conscious awareness is a separate issue.

>
>
> Whether or not you attack or defend the thesis, you can not escape this
>
> "hard" issue, as it's often called (iirc).
>
>
>
> > Are you familiar with the Turing test?
>
> >
>
> Yes. What about it here?
>

Well, I propose that we use something like the Turing test as our criterion for whether or not a given computer program is capable of adequately simulating a human mathematician (in response to your definitional question).

[Rupert]

On Friday, April 26, 2013 9:41:54 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 11:18 AM, Rupert wrote:
>
> > On Friday, April 26, 2013 6:59:30 PM UTC+2, Nam Nguyen wrote:
>
>
>
> >> (a) You don't know - at minimum up to this moment - and nobody knows
>
> >>
>
> >> what the "natural numbers", "standard structure [model]" be
>
> >>
>
> >> when there's a formula (e.g. cGC) that can't be asserted as true or
>
> >>
>
> >> false (and there are at least some good rationale that it's
>
> >>
>
> >> logically impossible to know so).
>
> >>
>
> >
>
> > There is no good rationale at all for your claim that it is logically impossible to know the truth-value of cGC.
>
>
>
> Considering last time we were on the issue, you couldn't defend,
>
> explain, what you would mean by a language structure being an
>
> incoherent concept, three's no rationale to make sense of your
>
> utterance above.
>

I don't recall making any claim that it was an incoherent concept. I personally understand the concept of structure perfectly well, as defined in say Shoenfield. I wasn't clear on what _you_ meant by "language structure". It is _your_ job to explain what you mean by it. And it is _your_ job to defend your claim that it is somehow impossible to know the truth-value of cGC. It is not my job to refute it. I am on quite solid ground in saying that you have not yet satisfactorily defended the claim.

> >
>
> > In any event, this is irrelevant.
>
>
>
> Of course it's relevant.

It's not. We are talking about a claim that a particular sentence is provable in a particular formal theory. This is in principle a machine-checkable matter. And I claim that on this occasion it can be verified by a feasible computation. What my position is on matters such as truth in the standard model of Peano Arithmetic is neither here nor there.

> The natural numbers with all of its "baggage"
>
> about arithmetic truth-relativity would be relevant to the purported
>
> consistency of PA we've been arguing about here and elsewhere.
>

It would not be relevant to the claim that I have made here, which is that a certain sentence, call it Con(PA), is provable in a certain formal theory, call it PA*. As I say, this claim simply isn't open to reasonable doubt.

>
>
> Didn't you once erroneously state that if PA is inconsistent, the
>
> language structure known "the natural numbers" is incoherent?
>

I doubt that I put it that way. I might have said that if PA is inconsistent then it would follow that we don't have a coherent conception of the natural numbers.

>
>
> > I only made remarks about what the "intended interpretation" of the unary predicate was to help your intuition about what axioms I wanted to postulate for it.
>
>
>
> But what's _your definition_ of "intended interpretation" and how would
>
> that help others to understand the "proof" that PA is consistent?
>

To understand what "intended interpretation" means, consult a dictionary. It shouldn't be too hard for anyone who knows the slightest bit of mathematical logic to understand the proof of my claim that a certain sentence Con(PA) is provable in a certain formal theory. You apparently struggle with it. I do not know how I can help you.

>
>
>
>
> > I could give a complete definition of the formal theory I want to talk about easily enough without making any reference at all to the notion of "truth in the standard model".
>
>
>
> > My claim that the consistency of PA is a theorem of this formal theory would then be a matter that is in principle machine-checkable.
>
>
>
> It's a trivial observation that a machine can check syntactical
>
> theorems, including those of the form F /\ ~F. How would you go
>
> from there to _claim_ the consistency of PA, something that per
>
> _definition of consistency_ is logically impossible to prove?
>

It is not logically impossible for Con(PA) to be provable in various formal theories; there are plenty of formal theories in which it is provable. I did not make any claim that PA is in fact consistent. If you trusted PA*, then you would conclude that PA is consistent. That is up to you.

>
>
> >
>
> >>
>
> >>
>
> >> (b) It's nonsensical to assert the consistency of PA outside
>
> >>
>
> >> the technical definition of formal system consistency which
>
> >>
>
> >> would involve unprovability _IN_ the underlying system,
>
> >>
>
> >> PA in this case.
>
> >>
>
> >
>
> > I have absolutely no idea what you are babbling on about. There is a certain sentence in the first-order language of arithmetic which we may call the consistency sentence for PA.
>
>
>
> Ah, so that's it! The consistency of PA would technically rest with what
>
> Rupert "may call the consistency sentence for PA"!
>

I don't know what you are babbling on about. The claim is not that PA is consistent. The claim is that the sentence Con(PA) is provable in a certain formal theory. If you trust the formal theory in question, then it would be reasonable to conclude that PA is consistent.

>
>
> And that would be called mathematical _logic_ ?
>
>
>
> > This sentence is a theorem of the formal theory which I was talking about. As I said, this can all be verified by a feasible computation.
>
>
>
> Sure. Giving me any sentence whatsoever and within seconds I'll come up
>
> with a formal theory that would prove it and of course can be "verified
>
> by a feasible computation" according to you.
>

Yes. But some claims about which sentences can be proved in which formal theories are more interesting than others.

In any event, I claim to be giving a correct paraphrase of the point that Torkel Franzen was making. Whether or not you find it to be of any interest is something about which I really couldn't care less. My claim was that it was unassailable.

>
>
> That's _easy_ .
>
> >
>
> >>
>
> >>
>
> >> Your alluded "In the resulting theory" isn't PA, hence it's not
>
> >>
>
> >> just insufficiency clear but also dubious as well as logically
>
> >>
>
> >> irrelevant as far as FOL _definition_ of formal system consistency.
>
> >>
>
> >
>
> > I have no idea what you're babbling on about.
>
> >
>
> > My claim is simply that a certain sentence is provable in a certain precisely defined recursively axiomatizable formal theory.
>
>
>
> I'm not babbling. I'm just telling you that you're bluffing in saying
>
> that the _real_ consistency of a formal system can be syntactically
>
> be proven by another formal system.
>

I have no idea what that is supposed to mean.

>
>
> It simply can't be.
>

Why not?

>
>
> >> Of course, if you don't follow FOL _definition_ of formal system
>
> >>
>
> >> consistency, it will be logically unclear what you were talking
>
> >>
>
> >> about in your "In the resulting theory, the consistency of PA is provable".
>
> >>
>
> >
>
> > I'm sorry you find it unclear. It's actually a perfectly precise and machine-checkable statement. I'm not sure whether I've got the energy to try to get you to understand.
>
>
>
> Your energy should be saved for your understanding that a "machine
>
> checkable" might guarantee an inconsistency but would have _NO_
>
> say in consistency.
>
>
>
> And that's a very simple observation in FOL reasoning that one
>
> should know.
>
>

There are certain circumstances under which it is reasonable to conclude that a particular formal theory is consistent. For example, PRA proves the consistency of Euclidean plane geometry. It is reasonable to conclude from this that Euclidean plane geometry is in fact consistent.

[Nam Nguyen]

On 26/04/2013 3:32 PM, Rupert wrote:
> On Friday, April 26, 2013 10:04:18 PM UTC+2, Nam Nguyen wrote:
>> On 26/04/2013 11:14 AM, Rupert wrote:
>>
>>> On Friday, April 26, 2013 6:36:06 PM UTC+2, Nam Nguyen wrote:
>>
>>>> On 26/04/2013 9:58 AM, Rupert wrote:
>>
>>
>>>>> On Friday, April 26, 2013 5:41:44 PM UTC+2, Nam Nguyen wrote:
>>
>>>>>> On 26/04/2013 4:04 AM, Rupert wrote:
>>
>>
>>> Do you or do you not believe that it is possible for a computer to simulate a human mathematician? Or do you have no opinion about the matter?
>>
>> Of course I do. My opinion is that Lucas-Penrose thesis is wrong.
>>
>> And I did explain why.
>>
>
> Okay, well thank you for clarifying that. You do understand that what you are saying here is that you do think that it is possible for a computer to simulate a human mathematician? That's your view?

Yes that's my view. The caveat is that, upon asked, I will have to be
really technical and present a technical definition of what is
mathematically meant "for a computer to simulate a human [mind]".

The question for you though is on your own, would you have such a
_technical definition_ ready for being presented, when asked?

>>
>>>> Iow, I was simply making a caveat to you that your "You have given
>>
>>>> no reason" shouldn't be referring to me.
>>
>>
>>> Right, I see. I apologize; I was confused between you and Newberry for a while.
>>
>>>> I did _NOT_ reveral my position on Lucas-Penrose thesis _there_ (on that
>>
>>>> statement you quoted).
>>
>>> Okay, so I misunderstood.
>>
>>
>> Then, iirc, your have the same opinion that Lucas-Penrose thesis is
>>
>> wrong. Right?
>>
>
> Well, yeah, I mean, I wouldn't claim to know whether or not it is in principle possible for a computer to simulate a human mathematician, but I would think that a whole brain emulation would probably do it, and I think that Lucas and Penrose don't adequtaely defend their view that it's not possible.

Agree, in general. To be more specific, the thesis is wrong in the
sense of being logically invalid because Lucas and Penrose made
a technical assertion ("the human mind is not a Turing machine")
without a clear technical definition of what the "human mind" be.

>>
>>>> > I just noticed that so far you and the other poster have not clearly,
>>>> > formally, meaningfully, soundly, defined what it'd mean for a
>>
>>>> > computer to have _perception_ the way a human does.
>>
>>>> See. Note my "the other poster" (who is Newberry).
>>
>>
>>> Yes, but my point stands that issues about "perception" have nothing to do with the Lucas-Penrose thesis.
>>
>> But _there_ you're wrong. Human intelligence, perception of
>>
>> mathematical truth, e.g., is the corner stone of the thesis.
>>
>
> Penrose claims that our capacity to "understand" mathematics plays an important role in our ability to do mathematics, and he believes that this is closely tied up with our capacity for conscious awareness. (What Lucas' views were I do not know.) Nevertheless, the thesis itself is just about the question of whether or not a computer could in principle simulate a human mathematician. That in itself has nothing to do with conscious awareness. Conscious awareness is a separate issue.
>
>> Whether or not you attack or defend the thesis, you can not escape this
>>
>> "hard" issue, as it's often called (iirc).
>>
>>> Are you familiar with the Turing test?
>>
>>
>> Yes. What about it here?
>>
>
> Well, I propose that we use something like the Turing test as our criterion for whether or not a given computer program is capable of adequately simulating a human mathematician (in response to your definitional question).

The Turing test is a good guidance but I'm looking for a more
technical definition, though. The last thing we'd want is for
the arguments to become philosophical, imho.

Could you give a more formal definition for what is meant by
a general machine (even abstract machine) simulating a human mind?

[Nam Nguyen]

In any rate, here's is my definition _in brief_ (assuming that
by a machine we'd mean the abstract Turing machine of which the
input and output would be infinite):

A machine capable of simulating a human mind is one in which
its output is _transcendental_ , when properly encoded as a real
number.

[fom]

On 4/24/2013 8:28 PM, Newberry wrote:
> So don't
> tell me that it is all based on manifest truth. In fact I have shown
> in another thread
> https://groups.google.com/forum/?hl=en&fromgroups#!topic/sci.logic/lDJcgOg4vco
> that the proof that the truths of first order arithmetic are not
> recursively enumerable is NOT likely to hold if we use Strawson-like
> semantics.
>

What exactly do you mean by "Strawson-like"
semantics. Did Strawson ever produce a semantic
theory?

[Rupert]

On Friday, April 26, 2013 11:57:10 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 3:32 PM, Rupert wrote:
>
> > On Friday, April 26, 2013 10:04:18 PM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 11:14 AM, Rupert wrote:
>
> >>
>
> >>> On Friday, April 26, 2013 6:36:06 PM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>>> On 26/04/2013 9:58 AM, Rupert wrote:
>
> >>
>
> >>
>
> >>>>> On Friday, April 26, 2013 5:41:44 PM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>>>>> On 26/04/2013 4:04 AM, Rupert wrote:
>
> >>
>
> >>
>
> >>> Do you or do you not believe that it is possible for a computer to simulate a human mathematician? Or do you have no opinion about the matter?
>
> >>
>
> >> Of course I do. My opinion is that Lucas-Penrose thesis is wrong.
>
> >>
>
> >> And I did explain why.
>
> >>
>
> >
>
> > Okay, well thank you for clarifying that. You do understand that what you are saying here is that you do think that it is possible for a computer to simulate a human mathematician? That's your view?
>
>
>
> Yes that's my view. The caveat is that, upon asked, I will have to be
>
> really technical and present a technical definition of what is
>
> mathematically meant "for a computer to simulate a human [mind]".
>
>
>
> The question for you though is on your own, would you have such a
>
> _technical definition_ ready for being presented, when asked?
>

I think it's silly to give a concept like that a "technical definition". I've given you my definition, in terms of the Turing test.

No, I can't. The issue is a philosophical issue. You can't expect a mathematical definition of the concepts involved.

[Rupert]

It is easy to prove that there exists such Turing machines (but it's a very silly definition).

[Nam Nguyen]

Why is that technical definition a silly one?

[Rupert]

On Saturday, April 27, 2013 7:33:30 AM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 10:51 PM, Rupert wrote:
>
> > On Saturday, April 27, 2013 1:13:20 AM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 3:57 PM, Nam Nguyen wrote:
>
>
>
> >>> Could you give a more formal definition for what is meant by
>
> >>> a general machine (even abstract machine) simulating a human mind?
>
> >>
>
> >> In any rate, here's is my definition _in brief_ (assuming that
>
> >> by a machine we'd mean the abstract Turing machine of which the
>
> >> input and output would be infinite):
>
> >>
>
> >> A machine capable of simulating a human mind is one in which
>
> >> its output is _transcendental_ , when properly encoded as a real
>
> >> number.
>
> >
>
> > It is easy to prove that there exists such Turing machines (but it's a very silly definition).
>
> >
>
> Why is that technical definition a silly one?
>

Because there obviously exists a Turing machine whose output is the binary expansion of e, and this is obviously not tantamount to a Turing machine simulating a human mathematician.

[Nam Nguyen]

So First Order definition of proof is silly because there's always that
silly proof of |- n=n which doesn't tantamount to a non-logical proof?

What about other non trivial cases of Turing machine?

In any rate, why would _that_ not be a simulation of transcendence of a
human mind?

[Rupert]

On Saturday, April 27, 2013 8:56:18 AM UTC+2, Nam Nguyen wrote:
> On 27/04/2013 12:25 AM, Rupert wrote:
>
> > On Saturday, April 27, 2013 7:33:30 AM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 10:51 PM, Rupert wrote:
>
> >>
>
> >>> On Saturday, April 27, 2013 1:13:20 AM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>>> On 26/04/2013 3:57 PM, Nam Nguyen wrote:
>
> >>
>
> >>
>
> >>
>
> >>>>> Could you give a more formal definition for what is meant by
>
> >>
>
> >>>>> a general machine (even abstract machine) simulating a human mind?
>
> >>
>
> >>>>
>
> >>
>
> >>>> In any rate, here's is my definition _in brief_ (assuming that
>
> >>
>
> >>>> by a machine we'd mean the abstract Turing machine of which the
>
> >>
>
> >>>> input and output would be infinite):
>
> >>
>
> >>>>
>
> >>
>
> >>>> A machine capable of simulating a human mind is one in which
>
> >>
>
> >>>> its output is _transcendental_ , when properly encoded as a real
>
> >>
>
> >>>> number.
>
> >>
>
> >>>
>
> >>
>
> >>> It is easy to prove that there exists such Turing machines (but it's a very silly definition).
>
> >>
>
> >>>
>
> >>
>
> >> Why is that technical definition a silly one?
>
> >>
>
> >
>
> > Because there obviously exists a Turing machine whose output is the binary expansion of e, and this is obviously not tantamount to a Turing machine simulating a human mathematician.
>
> >
>
> So First Order definition of proof is silly because there's always that
>
> silly proof of |- n=n which doesn't tantamount to a non-logical proof?
>

This is an ungrammatical sentence, and I have absolutely no idea what it is supposed to mean.

>
>
> What about other non trivial cases of Turing machine?
>

What about them?

>
>
> In any rate, why would _that_ not be a simulation of transcendence of a
>
> human mind?
>

Let me be absolutely sure that I understand. I can very easily write a computer program which will output the digits of e. You want me to explain to you why that is not tantamount to simulating a human mind?

[Nam Nguyen]

On 27/04/2013 1:28 AM, Rupert wrote:
> On Saturday, April 27, 2013 8:56:18 AM UTC+2, Nam Nguyen wrote:
>> On 27/04/2013 12:25 AM, Rupert wrote:
>>
>>> On Saturday, April 27, 2013 7:33:30 AM UTC+2, Nam Nguyen wrote:
>>
>>>> On 26/04/2013 10:51 PM, Rupert wrote:
>>
>>>>
>>
>>>>> On Saturday, April 27, 2013 1:13:20 AM UTC+2, Nam Nguyen wrote:
>>
>>>>
>>
>>>>>> On 26/04/2013 3:57 PM, Nam Nguyen wrote:
>>
>>>>
>>
>>>>
>>
>>>>
>>
>>>>>>> Could you give a more formal definition for what is meant by
>>
>>>>
>>
>>>>>>> a general machine (even abstract machine) simulating a human mind?
>>
>>>>
>>
>>>>>>
>>
>>>>
>>
>>>>>> In any rate, here's is my definition _in brief_ (assuming that
>>
>>>>
>>
>>>>>> by a machine we'd mean the abstract Turing machine of which the
>>
>>>>
>>
>>>>>> input and output would be infinite):
>>
>>>>
>>
>>>>>>
>>
>>>>
>>
>>>>>> A machine capable of simulating a human mind is one in which
>>
>>>>
>>
>>>>>> its output is _transcendental_ , when properly encoded as a real
>>
>>>>
>>
>>>>>> number.
>>
>>>>
>>
>>>>>
>>
>>>>
>>
>>>>> It is easy to prove that there exists such Turing machines (but it's a very silly definition).
>>
>>>>
>>
>>>>>
>>
>>>>
>>
>>>> Why is that technical definition a silly one?
>>
>>
>>> Because there obviously exists a Turing machine whose output is the binary expansion of e, and this is obviously not tantamount to a Turing machine simulating a human mathematician.
>>
>>>
>>
>> So First Order definition of proof is silly because there's always that
>>
>> silly proof of |- n=n which doesn't tantamount to a non-logical proof?
>>
>
> This is an ungrammatical sentence, and I have absolutely no idea what it is supposed to mean.

I was about to correct the grammar. But never mind. You don't seem to
care for what's behind the analogy and resort to grammar error, so let's
forget about the analogy, which is free and one doesn't have to take it.

>
>>
>>
>> What about other non trivial cases of Turing machine?
>>
>
> What about them?

Would they represent a simulation of human mind?

(Going this far, you are _unable_ to understand such a simple question?)

>>
>> In any rate, why would _that_ not be a simulation of transcendence of a
>> human mind?
>>
>
> Let me be absolutely sure that I understand.

Sure. Nothing wrong for asking for clarification.

> I can very easily write a computer program which will output the digits of e.

True. Any _human being_ with basic math and CS knowledge would be able
to do it.

But have you seen that ability from a randomly assembled machine or a
monkey?

> You want me to explain to you why that is not tantamount to simulating a human mind?

Yes please _definitely do explain_ to me and others why.

As I've asked you above, would you think a _randomly assembled_ machine
or a _monkey_ could "write a computer program which will output the
digits of e"?

Where could we find such a randomly assembled machine or such a monkey?

[Rupert]

I don't understand the analogy. Which proof of n=n did you have in mind?

> >
>
> >>
>
> >>
>
> >> What about other non trivial cases of Turing machine?
>
> >>
>
> >
>
> > What about them?
>
>
>
> Would they represent a simulation of human mind?
>

Well, it depends, doesn't it; I'd need to know more about them.

>
>
> (Going this far, you are _unable_ to understand such a simple question?)
>
>
>
> >>
>
> >> In any rate, why would _that_ not be a simulation of transcendence of a
>
> >> human mind?
>
> >>
>
> >
>
> > Let me be absolutely sure that I understand.
>
>
>
> Sure. Nothing wrong for asking for clarification.
>
>
>
> > I can very easily write a computer program which will output the digits of e.
>
>
>
> True. Any _human being_ with basic math and CS knowledge would be able
>
> to do it.
>
>
>
> But have you seen that ability from a randomly assembled machine or a
>
> monkey?
>

It would be unlikely.

>
>
> > You want me to explain to you why that is not tantamount to simulating a human mind?
>
>
>
> Yes please _definitely do explain_ to me and others why.
>

Because simulating a human mind would require processing inputs and producing outputs in a way that was in some way similar to what the human mind does.

>
>
> As I've asked you above, would you think a _randomly assembled_ machine
>
> or a _monkey_ could "write a computer program which will output the
>
> digits of e"?
>

It would be unlikely, but not totally out of the question.

>
>
> Where could we find such a randomly assembled machine or such a monkey?
>

I don't think that you're very likely to find one.

[Nam Nguyen]

On 26/04/2013 3:45 PM, Rupert wrote:
> On Friday, April 26, 2013 9:41:54 PM UTC+2, Nam Nguyen wrote:
>> On 26/04/2013 11:18 AM, Rupert wrote:
>>
>>> On Friday, April 26, 2013 6:59:30 PM UTC+2, Nam Nguyen wrote:
>>
>>>> (a) You don't know - at minimum up to this moment - and nobody knows
>>
>>>> what the "natural numbers", "standard structure [model]" be
>>>> when there's a formula (e.g. cGC) that can't be asserted as true or
>>>> false (and there are at least some good rationale that it's
>>>> logically impossible to know so).
>>
>>
>>> There is no good rationale at all for your claim that it is logically impossible to know the truth-value of cGC.
>>
>>
>>
>> Considering last time we were on the issue, you couldn't defend,
>>
>> explain, what you would mean by a language structure being an
>>
>> incoherent concept, three's no rationale to make sense of your
>>
>> utterance above.
>>
>
> I don't recall making any claim that it was an incoherent concept. I personally understand the concept of structure perfectly well, as defined in say Shoenfield.

Are you saying FOL has _many different definitions_ of language
structures: Shoenfield's, Enderton's etc... ?

(Gee, you sounded like the other poster!)

> I wasn't clear on what _you_ meant by "language structure".

It's a idiotic statement that you've made: you argued and understood
my many ... many .. examples of language structure (even finite ones)
before; and now you have to resort to your _pathetically lame excuse_
that you didn't know what I meant there as "language structure"!

No wonder those posts about mathematical relativity have gone nowhere:
your arguments aren't of in-good-faith nature!

> It is _your_ job to explain what you mean by it.

I _already did_ more than once. So it's _your job to notice_ that.

> And it is _your_ job to defend your claim that it is somehow impossible to know the truth-value of cGC.

I _already did_ more than once.

> It is not my job to refute it. I am on quite solid ground in saying that you have not yet satisfactorily defended the claim.

You're on a B.S. ground: you even flipped-flopped on you understanding
of the presented examples of language structures.

>>> I'm sorry you find it unclear. It's actually a perfectly precise and machine-checkable statement. I'm not sure whether I've got the energy to try to get you to understand.
>>
>> Your energy should be saved for your understanding that a "machine
>>
>> checkable" might guarantee an inconsistency but would have _NO_
>>
>> say in consistency.
>>
>> And that's a very simple observation in FOL reasoning that one
>>
>> should know.
>
> There are certain circumstances under which it is reasonable to conclude that a particular formal theory is consistent. For example, PRA proves the consistency of Euclidean plane geometry. It is reasonable to conclude from this that Euclidean plane geometry is in fact consistent.
>

Define what "reasonable" would technically mean.

[Rupert]

On Saturday, April 27, 2013 6:39:00 PM UTC+2, Nam Nguyen wrote:
> On 26/04/2013 3:45 PM, Rupert wrote:
>
> > On Friday, April 26, 2013 9:41:54 PM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 11:18 AM, Rupert wrote:
>
> >>
>
> >>> On Friday, April 26, 2013 6:59:30 PM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>>> (a) You don't know - at minimum up to this moment - and nobody knows
>
> >>
>
> >>>> what the "natural numbers", "standard structure [model]" be
>
> >>>> when there's a formula (e.g. cGC) that can't be asserted as true or
>
> >>>> false (and there are at least some good rationale that it's
>
> >>>> logically impossible to know so).
>
> >>
>
> >>
>
> >>> There is no good rationale at all for your claim that it is logically impossible to know the truth-value of cGC.
>
> >>
>
> >>
>
> >>
>
> >> Considering last time we were on the issue, you couldn't defend,
>
> >>
>
> >> explain, what you would mean by a language structure being an
>
> >>
>
> >> incoherent concept, three's no rationale to make sense of your
>
> >>
>
> >> utterance above.
>
> >>
>
> >
>
> > I don't recall making any claim that it was an incoherent concept. I personally understand the concept of structure perfectly well, as defined in say Shoenfield.
>
>
>
> Are you saying FOL has _many different definitions_ of language
>
> structures: Shoenfield's, Enderton's etc... ?
>

No.

>
>
> (Gee, you sounded like the other poster!)
>
>
>
> > I wasn't clear on what _you_ meant by "language structure".
>
>
>
> It's a idiotic statement that you've made: you argued and understood
>
> my many ... many .. examples of language structure (even finite ones)
>
> before; and now you have to resort to your _pathetically lame excuse_
>
> that you didn't know what I meant there as "language structure"!
>

Your insults strike me as unreasonable. Did you tell me what you meant by "language structure"? If you did, then just remind me. Is it the same as the definition in Shoenfield, for example? If you didn't tell me what you meant by the phrase, then it's not very reasonable to insult me for saying that I'm not clear on what you mean by it.

>
>
> No wonder those posts about mathematical relativity have gone nowhere:
>
> your arguments aren't of in-good-faith nature!
>

This belief of yours strikes me as without rational foundation.

>
>
> > It is _your_ job to explain what you mean by it.
>
>
>
> I _already did_ more than once. So it's _your job to notice_ that.
>

Did you really?

>
>
> > And it is _your_ job to defend your claim that it is somehow impossible to know the truth-value of cGC.
>
>
>
> I _already did_ more than once.
>

Not satisfactorily.

>
>
> > It is not my job to refute it. I am on quite solid ground in saying that you have not yet satisfactorily defended the claim.
>
>
>
> You're on a B.S. ground: you even flipped-flopped on you understanding
>
> of the presented examples of language structures.
>

You are wrong.

>
>
>
>
> >>> I'm sorry you find it unclear. It's actually a perfectly precise and machine-checkable statement. I'm not sure whether I've got the energy to try to get you to understand.
>
> >>
>
> >> Your energy should be saved for your understanding that a "machine
>
> >>
>
> >> checkable" might guarantee an inconsistency but would have _NO_
>
> >>
>
> >> say in consistency.
>
> >>
>
> >> And that's a very simple observation in FOL reasoning that one
>
> >>
>
> >> should know.
>
> >
>
> > There are certain circumstances under which it is reasonable to conclude that a particular formal theory is consistent. For example, PRA proves the consistency of Euclidean plane geometry. It is reasonable to conclude from this that Euclidean plane geometry is in fact consistent.
>
> >
>
>
>
> Define what "reasonable" would technically mean.
>

I suggest that you look the word up in a dictionary.

[Nam Nguyen]

On 27/04/2013 11:08 AM, Rupert wrote:
> On Saturday, April 27, 2013 6:39:00 PM UTC+2, Nam Nguyen wrote:
>> On 26/04/2013 3:45 PM, Rupert wrote:
>>
>> Are you saying FOL has _many different definitions_ of language
>>
>> structures: Shoenfield's, Enderton's etc... ?
>
> No.
>
>> (Gee, you sounded like the other poster!)
>>
>>> I wasn't clear on what _you_ meant by "language structure".
>>
>> It's a idiotic statement that you've made: you argued and understood
>>
>> my many ... many .. examples of language structure (even finite ones)
>>
>> before; and now you have to resort to your _pathetically lame excuse_
>>
>> that you didn't know what I meant there as "language structure"!
>>
>
> Your insults strike me as unreasonable. Did you tell me what you meant by "language structure"? If you did, then just remind me. Is it the same as the definition in Shoenfield, for example? If you didn't tell me what you meant by the phrase, then it's not very reasonable to insult me for saying that I'm not clear on what you mean by it.

For the record, Rupert, your paragraph above has just proven I'm correct
in complaining your argument style here is of the the nature of not-in
good-faith (as well as being idiotic).

As evident in this post:

https://groups.google.com/group/sci.logic/msg/60bbe19e38b2ff21?hl=en

Nam asked Rupert:

>> So, let me ask you this, to move ahead, would you understand
>> my Def1 in the case where the structure is finite?

For which Rupert replied:

> For finite structures, truth is decidable. This case is not
> interesting. It is the infinite case which is interesting.

So, _NOW AFTER 12+ MONTHS_ discussing about cGC you blamed me for
your not understanding what meant or would mean by language structure?

So, _why_ in the name of _being straightforward and being in-good-faith_
in a debate did you tell me:

- "For finite structures, truth is decidable"
- "This case is not interesting."
- "It is the infinite case which is interesting."

where "finite structures", "This case", "the infinite case", would
refer to my mentioned language structure?

You did claim to me that you are "a trained mathematician".
I would expect straightforwardness and in-good-faith in every
posters (including myself). But certainly from a trained mathematician!

So far, you haven't!

>>
>>> There are certain circumstances under which it is reasonable to conclude that a particular formal theory is consistent. For example, PRA proves the consistency of Euclidean plane geometry. It is reasonable to conclude from this that Euclidean plane geometry is in fact consistent.
>>
>>
>> Define what "reasonable" would technically mean.
>>
>
> I suggest that you look the word up in a dictionary.

Is that all a trained mathematician can use to _technically justify_
the "clarity" of the dubious phrase:

>>> There are certain circumstances under which it is reasonable to
>>> conclude that a particular formal theory is consistent.

given that FOL definition of consistency does _NOT_ grant any
_valid_ proof to conclude the consistency of a formal system?

[Nam Nguyen]

Let me emphasize that: " _my_ mentioned language structure?".

[Newberry]

On Thursday, April 25, 2013 10:40:24 PM UTC-7, Rupert wrote:
> On Friday, April 26, 2013 3:39:29 AM UTC+2, Newberry wrote:
>
> > On Thursday, April 25, 2013 9:27:31 AM UTC-7, Rupert wrote:
>
> >
>
> > > On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
>
> >
>
> >
>
> > > You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.
>
> >
>
> >
>
> >
>
> > So you know what a "proof" is after all. How can a computer believe in T'?
>
> >
>
>
>
> If the computer has cognitive states, then it can believe in T'. You may think a computer can't have cognitive states, but that's not relevant to assessing the Lucas-Penrose thesis. The Lucas-Penrose thesis is just about whether in principle a machine could be programmed to be behaviourally indistinguishable from a human mathematician.

How will the computer fathom this Platonic structure, which cannot be pinned down by a finite number of axioms?
>
>
>
> > > Well, obviously you have to assume some axioms if you want to prove anything at all.
>
> >
>
> >
>
> >
>
> > Do the axioms have to be true, are we talking about a meaningless game with symbols, or what are you saying here?
>
> >
>
>
>
> You'd be hoping that they were true.

Never lose hope!
>
>
>
> > > > How would you program a machine to perceive that the axioms of ZFC are
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > manifestly true?
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > What do you mean by that?
>
> >
>
> > Which word you did not understand?
>
> >
>
>
>
> "Perceive".

How will the computer acquire the knowledge that the axioms are manifestly true.
>
>
>
> > > > If you did a contradiction would result.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Why?
>
> >
>
> > By Gödel's second theorem.
>
> >
>
>
>
> All that Gödel's second theorem tells you is that if ZFC is consistent, then there is no proof in ZFC that ZFC is consistent. But there is no good reason to suppose that the things that the machine can "perceive" are limited to what can be proved in ZFC.
>
>
>
> If there were some formal system F which encoded the insights available to the machine and F were a consistent extension of ZFC, then the machine would be unable to perceive the consistency of F.

Yes, but Franzen's proof purports to be absolute.
>
>
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> > >
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> > >
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> >
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> > > > BTW, my main thesis was that what Franzen says is not true.
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> >
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> > >
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> >
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> > >
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> >
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> > >
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> >
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> > > Which thing that he says is not true?
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> >
>
> > My thesis is that the following argument by Torkel Franzen is incorrect:
>
> >
>
> > The axioms of ZFC are manifestly true. The rules of logic are truth preserving. Therefore everything we derive by those rules from those axioms is true. If everything is true then there is no contradiction. Hence ZFC is consistent.
>
> >
>
> > [BTW, I am NOT claiming that ZFC is not consistent.]
>
>
>
> Do you mean that the argument is not valid, or that one of its premises is false? Which one?

The argument is not valid. The logical apparatus injects spurious, extra-axiomatic "truths".

[Rupert]

On Sunday, April 28, 2013 1:08:22 AM UTC+2, Newberry wrote:
> On Thursday, April 25, 2013 10:40:24 PM UTC-7, Rupert wrote:
>
> > On Friday, April 26, 2013 3:39:29 AM UTC+2, Newberry wrote:
>
> >
>
> > > On Thursday, April 25, 2013 9:27:31 AM UTC-7, Rupert wrote:
>
> >
>
> > >
>
> >
>
> > > > On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.
>
> >
>
> > >
>
> >
>
> > >
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> >
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> > >
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> >
>
> > > So you know what a "proof" is after all. How can a computer believe in T'?
>
> >
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> > >
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> >
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> >
>
> >
>
> > If the computer has cognitive states, then it can believe in T'. You may think a computer can't have cognitive states, but that's not relevant to assessing the Lucas-Penrose thesis. The Lucas-Penrose thesis is just about whether in principle a machine could be programmed to be behaviourally indistinguishable from a human mathematician.
>
>
>
> How will the computer fathom this Platonic structure, which cannot be pinned down by a finite number of axioms?
>

Which Platonic structure?

How do you suppose that a human mathematician can do it?

> >
>
> >
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> >
>
> > > > Well, obviously you have to assume some axioms if you want to prove anything at all.
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> >
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> > >
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> >
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> > >
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> >
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> > >
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> >
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> > > Do the axioms have to be true, are we talking about a meaningless game with symbols, or what are you saying here?
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> >
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> > >
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> >
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> >
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> >
>
> > You'd be hoping that they were true.
>
>
>
> Never lose hope!
>
> >
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> >
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> >
>
> > > > > How would you program a machine to perceive that the axioms of ZFC are
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> >
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> > >
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> >
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> > > >
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> >
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> > >
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> >
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> > > > >
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> > >
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> > > >
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> >
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> > >
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> >
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> > > > > manifestly true?
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> >
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> > >
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> > >
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> >
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> > >
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> >
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> > > > What do you mean by that?
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> >
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> > >
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> >
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> > > Which word you did not understand?
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> >
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> > >
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> >
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> >
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> >
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> > "Perceive".
>
>
>
> How will the computer acquire the knowledge that the axioms are manifestly true.
>

I don't know. We haven't solved the problem of strong AI as yet. We don't know exactly what form it will take. You could imagine that we might construct a whole brain emulation. Then the mathematical education of the AI will be just like that of a human mathematician, and it will come to feel confident that the axioms of ZFC are manifestly true by whatever route a human mathematician comes to feel confident about that (and we don't know much about that, and possibly still wouldn't know much about it even if we succeeded in constructing a whole brain emulation). Or perhaps strong AI will be developed along other lines, and the confidence will come about by some other means. We don't know.

> >
>
> >
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> >
>
> > > > > If you did a contradiction would result.
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> >
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> > >
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> > >
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> >
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> > > > Why?
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> >
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> > >
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> >
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> > > By Gödel's second theorem.
>
> >
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> > >
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> >
>
> >
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> >
>
> > All that Gödel's second theorem tells you is that if ZFC is consistent, then there is no proof in ZFC that ZFC is consistent. But there is no good reason to suppose that the things that the machine can "perceive" are limited to what can be proved in ZFC.
>
> >
>
> >
>
> >
>
> > If there were some formal system F which encoded the insights available to the machine and F were a consistent extension of ZFC, then the machine would be unable to perceive the consistency of F.
>
>
>
> Yes, but Franzen's proof purports to be absolute.
>

His proof of what?

> >
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> > > > > BTW, my main thesis was that what Franzen says is not true.
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> > >
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> >
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> > > > Which thing that he says is not true?
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> >
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> > >
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> >
>
> > > My thesis is that the following argument by Torkel Franzen is incorrect:
>
> >
>
> > >
>
> >
>
> > > The axioms of ZFC are manifestly true. The rules of logic are truth preserving. Therefore everything we derive by those rules from those axioms is true. If everything is true then there is no contradiction. Hence ZFC is consistent.
>
> >
>
> > >
>
> >
>
> > > [BTW, I am NOT claiming that ZFC is not consistent.]
>
> >
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> >
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> >
>
> > Do you mean that the argument is not valid, or that one of its premises is false? Which one?
>
>
>
> The argument is not valid. The logical apparatus injects spurious, extra-axiomatic "truths".

Let's write the argument down step by step:

(1) The axioms of ZFC are manifestly true [Premise 1]
(2) The rules of logic are truth preserving [Premise 2]
(3) Therefore everything we derive by those rules from those axioms is true [from (1) and (2)]
(4) If everything is true then there is no contradiction [Premise 3]
(5) Hence ZFC is consistent [from (3) and (4)]

Now, do you know what "valid" means? You can reject Premise 1, Premise 2, or Premise 3 if you like. That is the option I gave you when I said you could reject one of the premises. But if you are prepared to grant all three premises, are you really maintaining that the conclusion does not logically follow?

It sounds as though you reject the notion of set-theoretic "truth" and therefore would reject Premise 1. Would that be a fair paraphrase?

[Rupert]

On Saturday, April 27, 2013 10:33:38 PM UTC+2, Nam Nguyen wrote:
> On 27/04/2013 11:08 AM, Rupert wrote:
>
> > On Saturday, April 27, 2013 6:39:00 PM UTC+2, Nam Nguyen wrote:
>
> >> On 26/04/2013 3:45 PM, Rupert wrote:
>
> >>
>
> >> Are you saying FOL has _many different definitions_ of language
>
> >>
>
> >> structures: Shoenfield's, Enderton's etc... ?
>
> >
>
> > No.
>
> >
>
> >> (Gee, you sounded like the other poster!)
>
> >>
>
> >>> I wasn't clear on what _you_ meant by "language structure".
>
> >>
>
> >> It's a idiotic statement that you've made: you argued and understood
>
> >>
>
> >> my many ... many .. examples of language structure (even finite ones)
>
> >>
>
> >> before; and now you have to resort to your _pathetically lame excuse_
>
> >>
>
> >> that you didn't know what I meant there as "language structure"!
>
> >>
>
> >
>
> > Your insults strike me as unreasonable. Did you tell me what you meant by "language structure"? If you did, then just remind me. Is it the same as the definition in Shoenfield, for example? If you didn't tell me what you meant by the phrase, then it's not very reasonable to insult me for saying that I'm not clear on what you mean by it.
>
>
>
> For the record, Rupert, your paragraph above has just proven I'm correct
>
> in complaining your argument style here is of the the nature of not-in
>
> good-faith (as well as being idiotic).
>

Well, that's a very insulting thing to say, and it strikes me as being totally without rational foundation. I think you are being unreasonable.

>
>
> As evident in this post:
>
>
>
> https://groups.google.com/group/sci.logic/msg/60bbe19e38b2ff21?hl=en
>
>
>
> Nam asked Rupert:
>
>
>
> >> So, let me ask you this, to move ahead, would you understand
>
> >> my Def1 in the case where the structure is finite?
>
>
>
> For which Rupert replied:
>
>
>
> > For finite structures, truth is decidable. This case is not
>
> > interesting. It is the infinite case which is interesting.
>
>
>
> So, _NOW AFTER 12+ MONTHS_ discussing about cGC you blamed me for
>
> your not understanding what meant or would mean by language structure?
>

Yes.

>
>
> So, _why_ in the name of _being straightforward and being in-good-faith_
>
> in a debate did you tell me:
>
>
>
> - "For finite structures, truth is decidable"
>
> - "This case is not interesting."
>
> - "It is the infinite case which is interesting."
>

Because that's true.

>
>
> where "finite structures", "This case", "the infinite case", would
>
> refer to my mentioned language structure?
>

If you mean by "language structure" just a structure in Shoenfield's sense, then that's fine. You only had to say so. That's what I meant by "structure" in that context.

>
>
> You did claim to me that you are "a trained mathematician".
>
> I would expect straightforwardness and in-good-faith in every
>
> posters (including myself). But certainly from a trained mathematician!
>

I am being straightforward and arguing in good faith. You have no good reason to suppose otherwise. Your accusations to the contrary are unreasonable.

>
>
> So far, you haven't!
>

No, that's not true. You're being unreasonable.

>
>
> >>
>
> >>> There are certain circumstances under which it is reasonable to conclude that a particular formal theory is consistent. For example, PRA proves the consistency of Euclidean plane geometry. It is reasonable to conclude from this that Euclidean plane geometry is in fact consistent.
>
> >>
>
> >>
>
> >> Define what "reasonable" would technically mean.
>
> >>
>
> >
>
> > I suggest that you look the word up in a dictionary.
>
>
>
>
>
> Is that all a trained mathematician can use to _technically justify_
>
> the "clarity" of the dubious phrase:
>
>
>
> >>> There are certain circumstances under which it is reasonable to
>
> >>> conclude that a particular formal theory is consistent.
>

The term "reasonable" in this context is not one which has a mathematical definition. Being a trained mathematician is neither here nor there when it comes to understanding this word. Native speakers of English could reasonably be expected to understand the word.

>
>
> given that FOL definition of consistency does _NOT_ grant any
>
> _valid_ proof to conclude the consistency of a formal system?
>

It does. For example, the consistency of Euclidean plane geometry can be proved in PRA. The consistency of Q can also be proved in PRA, and you can read about that in your favourite book by Shoenfield.

[Newberry]

On Saturday, April 27, 2013 4:24:09 PM UTC-7, Rupert wrote:
> On Sunday, April 28, 2013 1:08:22 AM UTC+2, Newberry wrote:
>
> > On Thursday, April 25, 2013 10:40:24 PM UTC-7, Rupert wrote:
>
> >
>
> > > On Friday, April 26, 2013 3:39:29 AM UTC+2, Newberry wrote:
>
> >
>
> > >
>
> >
>
> > > > On Thursday, April 25, 2013 9:27:31 AM UTC-7, Rupert wrote:
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
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> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > So you know what a "proof" is after all. How can a computer believe in T'?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
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> > >
>
> >
>
> > > If the computer has cognitive states, then it can believe in T'. You may think a computer can't have cognitive states, but that's not relevant to assessing the Lucas-Penrose thesis. The Lucas-Penrose thesis is just about whether in principle a machine could be programmed to be behaviourally indistinguishable from a human mathematician.
>
> >
>
> >
>
> >
>
> > How will the computer fathom this Platonic structure, which cannot be pinned down by a finite number of axioms?
>
> >
>
>
>
> Which Platonic structure?

The natural numbers. Why don't you stop playing dumb?

>
>
>
> How do you suppose that a human mathematician can do it?
>
>
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > > Well, obviously you have to assume some axioms if you want to prove anything at all.
>
> >
>
> > >
>
> >
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> > > >
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> >
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> > >
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>
> >
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> > >
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> >
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> > > >
>
> >
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> > >
>
> >
>
> > > > Do the axioms have to be true, are we talking about a meaningless game with symbols, or what are you saying here?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
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> > >
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> >
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> > >
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> >
>
> > >
>
> >
>
> > > You'd be hoping that they were true.
>
> >
>
> >
>
> >
>
> > Never lose hope!
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > > > How would you program a machine to perceive that the axioms of ZFC are
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
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> >
>
> > > > >
>
> >
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> > >
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> >
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> > > >
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> >
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> > >
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> >
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> > > > > >
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> >
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> > >
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> >
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> > >
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> >
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> >
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> > >
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> >
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> > > >
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> >
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> > >
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> >
>
> > > > > > manifestly true?
>
> >
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> > >
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> >
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> > > >
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> >
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> > >
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> >
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> > > > >
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> >
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> > >
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> > >
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> > > > >
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> >
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> > >
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> > >
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> >
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> > > > >
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> >
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> > >
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> >
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> > > >
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> >
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> > >
>
> >
>
> > > > > What do you mean by that?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > Which word you did not understand?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > "Perceive".
>
> >
>
> >
>
> >
>
> > How will the computer acquire the knowledge that the axioms are manifestly true.
>
> >
>
>
>
> I don't know. We haven't solved the problem of strong AI as yet. We don't know exactly what form it will take. You could imagine that we might construct a whole brain emulation. Then the mathematical education of the AI will be just like that of a human mathematician, and it will come to feel confident that the axioms of ZFC are manifestly true by whatever route a human mathematician comes to feel confident about that (and we don't know much about that, and possibly still wouldn't know much about it even if we succeeded in constructing a whole brain emulation). Or perhaps strong AI will be developed along other lines, and the confidence will come about by some other means. We don't know.

Goedel's second theorem indicates that we cannot solve it.

>
>
>
> > >
>
> >
>
> > >
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> >
>
> > >
>
> >
>
> > > > > > If you did a contradiction would result.
>
> >
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> > >
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> > > >
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> > >
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> > > > >
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> > > > > >
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> > >
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> > >
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> > > > >
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> > >
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> > >
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> > > >
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> > >
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>
> >
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> > >
>
> >
>
> > > >
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> >
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> > >
>
> >
>
> > > > > Why?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > By Gödel's second theorem.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > All that Gödel's second theorem tells you is that if ZFC is consistent, then there is no proof in ZFC that ZFC is consistent. But there is no good reason to suppose that the things that the machine can "perceive" are limited to what can be proved in ZFC.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > If there were some formal system F which encoded the insights available to the machine and F were a consistent extension of ZFC, then the machine would be unable to perceive the consistency of F.
>
> >
>
> >
>
> >
>
> > Yes, but Franzen's proof purports to be absolute.
>
> >
>
>
>
> His proof of what?

The proof we have been talking all along - that ZFC is consistent. WHy don't you stop playing dumb?

>
>
>
> > >
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> >
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> > > >
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> >
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> > >
>
> >
>
> > > > > > BTW, my main thesis was that what Franzen says is not true.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
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> > > > > Which thing that he says is not true?
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> > > > My thesis is that the following argument by Torkel Franzen is incorrect:
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> > > > The axioms of ZFC are manifestly true. The rules of logic are truth preserving. Therefore everything we derive by those rules from those axioms is true. If everything is true then there is no contradiction. Hence ZFC is consistent.
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> > > > [BTW, I am NOT claiming that ZFC is not consistent.]
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> > > Do you mean that the argument is not valid, or that one of its premises is false? Which one?
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> > The argument is not valid. The logical apparatus injects spurious, extra-axiomatic "truths".
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> Let's write the argument down step by step:
>

This proof!

>
>
> (1) The axioms of ZFC are manifestly true [Premise 1]
>
> (2) The rules of logic are truth preserving [Premise 2]
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> (3) Therefore everything we derive by those rules from those axioms is true [from (1) and (2)]
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> (4) If everything is true then there is no contradiction [Premise 3]
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> (5) Hence ZFC is consistent [from (3) and (4)]
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>
>
> Now, do you know what "valid" means? You can reject Premise 1, Premise 2, or Premise 3 if you like. That is the option I gave you when I said you could reject one of the premises. But if you are prepared to grant all three premises, are you really maintaining that the conclusion does not logically follow?
>

Well, the error is subtle. That is my whole point. I interpreted "truth preserving" as not losing any truth. (Maybe it is the long interpretation.) Anyway, the point is that there are some extra-axiomatic truths in classical logic i.e. tautologies, validities. Franzen did not examine if these are also manifestly true. In fact some of them are not and that is the error in his argument.

I guess now you are going to ask me which argument.

[Rupert]

On Sunday, April 28, 2013 5:30:13 AM UTC+2, Newberry wrote:
> On Saturday, April 27, 2013 4:24:09 PM UTC-7, Rupert wrote:
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> > On Sunday, April 28, 2013 1:08:22 AM UTC+2, Newberry wrote:
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> > > On Thursday, April 25, 2013 10:40:24 PM UTC-7, Rupert wrote:
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> > > > On Friday, April 26, 2013 3:39:29 AM UTC+2, Newberry wrote:
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> > > > > On Thursday, April 25, 2013 9:27:31 AM UTC-7, Rupert wrote:
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> > > > > > On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
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> > > > > > You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.
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> > > > > So you know what a "proof" is after all. How can a computer believe in T'?
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> > > > If the computer has cognitive states, then it can believe in T'. You may think a computer can't have cognitive states, but that's not relevant to assessing the Lucas-Penrose thesis. The Lucas-Penrose thesis is just about whether in principle a machine could be programmed to be behaviourally indistinguishable from a human mathematician.
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> > > How will the computer fathom this Platonic structure, which cannot be pinned down by a finite number of axioms?
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> > Which Platonic structure?
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> The natural numbers. Why don't you stop playing dumb?
>

I wasn't playing dumb. The context was not enough for me to know the answer to your question. Please do not be rude.

Somehow or other a human mathematician is able to engage in cognition about the natural numbers. There is no good reason to suppose that the brain processes that occur when this happens could not be adequately simulated by a computer.

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> > > > > Do the axioms have to be true, are we talking about a meaningless game with symbols, or what are you saying here?
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> > > > > > > How would you program a machine to perceive that the axioms of ZFC are
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> > > > > > > manifestly true?
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> > > > > > What do you mean by that?
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> > > > > Which word you did not understand?
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> > > > "Perceive".
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> > > How will the computer acquire the knowledge that the axioms are manifestly true.
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> > I don't know. We haven't solved the problem of strong AI as yet. We don't know exactly what form it will take. You could imagine that we might construct a whole brain emulation. Then the mathematical education of the AI will be just like that of a human mathematician, and it will come to feel confident that the axioms of ZFC are manifestly true by whatever route a human mathematician comes to feel confident about that (and we don't know much about that, and possibly still wouldn't know much about it even if we succeeded in constructing a whole brain emulation). Or perhaps strong AI will be developed along other lines, and the confidence will come about by some other means. We don't know.
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> Goedel's second theorem indicates that we cannot solve it.
>

That's false.

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> > > > > > > If you did a contradiction would result.
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> > > > > > Why?
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> > > > > By Gödel's second theorem.
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> > > > All that Gödel's second theorem tells you is that if ZFC is consistent, then there is no proof in ZFC that ZFC is consistent. But there is no good reason to suppose that the things that the machine can "perceive" are limited to what can be proved in ZFC.
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> > > > If there were some formal system F which encoded the insights available to the machine and F were a consistent extension of ZFC, then the machine would be unable to perceive the consistency of F.
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> > > Yes, but Franzen's proof purports to be absolute.
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> > His proof of what?
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> The proof we have been talking all along - that ZFC is consistent. WHy don't you stop playing dumb?

Let's say that ZFC is a theory in the first-order language of set theory, and that this language has a denumerable list of individual variables v_0, v_1, v_2, ...

Add a truth predicate True(x) to the language of ZFC. The intended interpretation is that True(x) will be true when x is an ordered pair whose first term is a Goedel code for a well-formed formula in the first-order language of set theory and whose second term is a finite sequence (x_0, x_1, ... x_n) such that all the free variables of the well-formed formula are among v_0, v_1, v_2, ... v_n, and the formula is true under any interpretation which assigns x_0 to v_0, x_1 to v_1, ... x_n to v_n.

Add to ZFC the obvious axioms governing the truth predicate which correspond to the Tarski clauses. Extend the axiom schema of Separation and the axiom schema of Replacement to include formulas which contain the truth predicate. Call the resulting theory ZFC*.

Franzen's proof can be formalized in ZFC*. This is not a counterexample to Goedel's second theorem. Goedel's second theorem applies to ZFC*. If ZFC* is consistent, then ZFC* cannot prove the consistency of ZFC*. But it can prove the consistency of ZFC.

So what's the problem?

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> > > > > > > BTW, my main thesis was that what Franzen says is not true.
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> > > > > > Which thing that he says is not true?
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> > > > > My thesis is that the following argument by Torkel Franzen is incorrect:
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> > > > > The axioms of ZFC are manifestly true. The rules of logic are truth preserving. Therefore everything we derive by those rules from those axioms is true. If everything is true then there is no contradiction. Hence ZFC is consistent.
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> > > > > [BTW, I am NOT claiming that ZFC is not consistent.]
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> > > > Do you mean that the argument is not valid, or that one of its premises is false? Which one?
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> > > The argument is not valid. The logical apparatus injects spurious, extra-axiomatic "truths".
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> > Let's write the argument down step by step:
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> >
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> This proof!
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> >
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> >
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> > (1) The axioms of ZFC are manifestly true [Premise 1]
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> >
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> > (2) The rules of logic are truth preserving [Premise 2]
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> >
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> > (3) Therefore everything we derive by those rules from those axioms is true [from (1) and (2)]
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> >
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> > (4) If everything is true then there is no contradiction [Premise 3]
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> >
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> > (5) Hence ZFC is consistent [from (3) and (4)]
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> >
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> >
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> >
>
> > Now, do you know what "valid" means? You can reject Premise 1, Premise 2, or Premise 3 if you like. That is the option I gave you when I said you could reject one of the premises. But if you are prepared to grant all three premises, are you really maintaining that the conclusion does not logically follow?
>
> >
>
> Well, the error is subtle. That is my whole point. I interpreted "truth preserving" as not losing any truth. (Maybe it is the long interpretation.) Anyway, the point is that there are some extra-axiomatic truths in classical logic i.e. tautologies, validities. Franzen did not examine if these are also manifestly true. In fact some of them are not and that is the error in his argument.
>

Right, so what is an example of a formula in the first-order language of set theory which is provable in the classical predicate calculus but is not manifestly true?

[Newberry]

On Saturday, April 27, 2013 9:13:06 PM UTC-7, Rupert wrote:
> On Sunday, April 28, 2013 5:30:13 AM UTC+2, Newberry wrote:
>
> > On Saturday, April 27, 2013 4:24:09 PM UTC-7, Rupert wrote:
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> >
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> > > On Sunday, April 28, 2013 1:08:22 AM UTC+2, Newberry wrote:
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> >
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> > >
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> >
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> > > > On Thursday, April 25, 2013 10:40:24 PM UTC-7, Rupert wrote:
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> > >
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> >
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> > > >
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> > > > > On Friday, April 26, 2013 3:39:29 AM UTC+2, Newberry wrote:
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> > > > > > On Thursday, April 25, 2013 9:27:31 AM UTC-7, Rupert wrote:
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> > > > > > > On Thursday, April 25, 2013 3:39:44 PM UTC+2, Newberry wrote:
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> > > > > > > You have a proof that T is consistent (in T') no matter what, but whether you find this proof convincing will depend whether you believe in T'.
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> > > > > > So you know what a "proof" is after all. How can a computer believe in T'?
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> > > > > If the computer has cognitive states, then it can believe in T'. You may think a computer can't have cognitive states, but that's not relevant to assessing the Lucas-Penrose thesis. The Lucas-Penrose thesis is just about whether in principle a machine could be programmed to be behaviourally indistinguishable from a human mathematician.
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> > > > How will the computer fathom this Platonic structure, which cannot be pinned down by a finite number of axioms?
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> > > Which Platonic structure?
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> > The natural numbers. Why don't you stop playing dumb?
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>
> I wasn't playing dumb. The context was not enough for me to know the answer to your question. Please do not be rude.
>
>
>
> Somehow or other a human mathematician is able to engage in cognition about the natural numbers. There is no good reason to suppose that the brain processes that occur when this happens could not be adequately simulated by a computer.

How can it be simulated if it cannot be captured by a finite number of axioms or axiom schemata?

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> > > How do you suppose that a human mathematician can do it?
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> > > > > > > Well, obviously you have to assume some axioms if you want to prove anything at all.
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> > > > > > Do the axioms have to be true, are we talking about a meaningless game with symbols, or what are you saying here?
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> > > > > You'd be hoping that they were true.
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> > > > Never lose hope!
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> > > > > > > > How would you program a machine to perceive that the axioms of ZFC are
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> > > > > > > > manifestly true?
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> > > > > > > What do you mean by that?
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> > > > > > Which word you did not understand?
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> > > > > "Perceive".
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> > > > How will the computer acquire the knowledge that the axioms are manifestly true.
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> > > I don't know. We haven't solved the problem of strong AI as yet. We don't know exactly what form it will take. You could imagine that we might construct a whole brain emulation. Then the mathematical education of the AI will be just like that of a human mathematician, and it will come to feel confident that the axioms of ZFC are manifestly true by whatever route a human mathematician comes to feel confident about that (and we don't know much about that, and possibly still wouldn't know much about it even if we succeeded in constructing a whole brain emulation). Or perhaps strong AI will be developed along other lines, and the confidence will come about by some other means. We don't know.
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> > Goedel's second theorem indicates that we cannot solve it.
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> That's false.

Here is your contradiction. Franzen's proof purports to be absolute proof of ZFC consistency. Gödel's second theorem indicates that you can do so only in an infinite hierarchy of formal systems. Franzen somehow collapsed all the hierarchy into a few sentences.

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> > > > > > > > If you did a contradiction would result.
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> > > > > > > Why?
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> > > > > > By Gödel's second theorem.
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> > > > > All that Gödel's second theorem tells you is that if ZFC is consistent, then there is no proof in ZFC that ZFC is consistent. But there is no good reason to suppose that the things that the machine can "perceive" are limited to what can be proved in ZFC.
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> > > > > If there were some formal system F which encoded the insights available to the machine and F were a consistent extension of ZFC, then the machine would be unable to perceive the consistency of F.
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> > > > Yes, but Franzen's proof purports to be absolute.
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> > > His proof of what?
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> > The proof we have been talking all along - that ZFC is consistent. WHy don't you stop playing dumb?
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> Let's say that ZFC is a theory in the first-order language of set theory, and that this language has a denumerable list of individual variables v_0, v_1, v_2, ...
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> Add a truth predicate True(x) to the language of ZFC. The intended interpretation is that True(x) will be true when x is an ordered pair whose first term is a Goedel code for a well-formed formula in the first-order language of set theory and whose second term is a finite sequence (x_0, x_1, ... x_n) such that all the free variables of the well-formed formula are among v_0, v_1, v_2, ... v_n, and the formula is true under any interpretation which assigns x_0 to v_0, x_1 to v_1, ... x_n to v_n.
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> Add to ZFC the obvious axioms governing the truth predicate which correspond to the Tarski clauses. Extend the axiom schema of Separation and the axiom schema of Replacement to include formulas which contain the truth predicate. Call the resulting theory ZFC*.
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> Franzen's proof can be formalized in ZFC*. This is not a counterexample to Goedel's second theorem. Goedel's second theorem applies to ZFC*. If ZFC* is consistent, then ZFC* cannot prove the consistency of ZFC*. But it can prove the consistency of ZFC.
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> So what's the problem?

The problem is that this is NOT Franzen;s proof. (Firstly Franzen's proof is about consistency of ZFC not PA, but let's not dwell on this.) But mainly Franzen's proof is not relative to any other system whose consistency is unknown. Franzen's proof is absolute based on manifest truth.

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> > > > > > > > BTW, my main thesis was that what Franzen says is not true.
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> > > > > > > Which thing that he says is not true?
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> > > > > > My thesis is that the following argument by Torkel Franzen is incorrect:
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> > > > > > The axioms of ZFC are manifestly true. The rules of logic are truth preserving. Therefore everything we derive by those rules from those axioms is true. If everything is true then there is no contradiction. Hence ZFC is consistent.
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> > > > > > [BTW, I am NOT claiming that ZFC is not consistent.]
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> >
>
> > > > > Do you mean that the argument is not valid, or that one of its premises is false? Which one?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > The argument is not valid. The logical apparatus injects spurious, extra-axiomatic "truths".
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Let's write the argument down step by step:
>
> >
>
> > >
>
> >
>
> > This proof!
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > (1) The axioms of ZFC are manifestly true [Premise 1]
>
> >
>
> > >
>
> >
>
> > > (2) The rules of logic are truth preserving [Premise 2]
>
> >
>
> > >
>
> >
>
> > > (3) Therefore everything we derive by those rules from those axioms is true [from (1) and (2)]
>
> >
>
> > >
>
> >
>
> > > (4) If everything is true then there is no contradiction [Premise 3]
>
> >
>
> > >
>
> >
>
> > > (5) Hence ZFC is consistent [from (3) and (4)]
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Now, do you know what "valid" means? You can reject Premise 1, Premise 2, or Premise 3 if you like. That is the option I gave you when I said you could reject one of the premises. But if you are prepared to grant all three premises, are you really maintaining that the conclusion does not logically follow?
>
> >
>
> > >
>
> >
>
> > Well, the error is subtle. That is my whole point. I interpreted "truth preserving" as not losing any truth. (Maybe it is the long interpretation.) Anyway, the point is that there are some extra-axiomatic truths in classical logic i.e. tautologies, validities. Franzen did not examine if these are also manifestly true. In fact some of them are not and that is the error in his argument.
>
> >
>
>
>
> Right, so what is an example of a formula in the first-order language of set theory which is provable in the classical predicate calculus but is not manifestly true?

Example 1:
(P & ~P) -> Q

Example 2:
(x)((x+3 < x) -> (x = x+4))

If you want to argue from manifest truth then the paradox of material implication immediately becomes a suspect.

[Nam Nguyen]

On 27/04/2013 5:31 PM, Rupert wrote:
> On Saturday, April 27, 2013 10:33:38 PM UTC+2, Nam Nguyen wrote:

>>
>> For the record, Rupert, your paragraph above has just proven I'm correct
>>
>> in complaining your argument style here is of the the nature of not-in
>>
>> good-faith (as well as being idiotic).
>>
>
> Well, that's a very insulting thing to say, and it strikes me as being totally without rational foundation. I think you are being unreasonable.

I don't accuse you without evidences (facts) to back it up. See more
below.

>> As evident in this post:
>>
>> https://groups.google.com/group/sci.logic/msg/60bbe19e38b2ff21?hl=en
>>
>> Nam asked Rupert:
>>
>> >> So, let me ask you this, to move ahead, would you understand
>>
>> >> my Def1 in the case where the structure is finite?
>>
>> For which Rupert replied:
>>
>> > For finite structures, truth is decidable. This case is not
>> > interesting. It is the infinite case which is interesting.
>>
>> So, _NOW AFTER 12+ MONTHS_ discussing about cGC you blamed me for
>> your not understanding what meant or would mean by language structure?
>>
> Yes.
>
>> So, _why_ in the name of _being straightforward and being in-good-faith_
>> in a debate did you tell me:
>>
>> - "For finite structures, truth is decidable"
>>
>> - "This case is not interesting."
>>
>> - "It is the infinite case which is interesting."
>>
>
> Because that's true.
>
>>
>> where "finite structures", "This case", "the infinite case", would
>>
>> refer to my mentioned language structure?
>>
>
> If you mean by "language structure" just a structure in Shoenfield's sense, then that's fine. You only had to say so. That's what I meant by "structure" in that context.

I already did. In that thread I specifically said to you I was referring
to page 19. of Section 2.5 "Structures" when I talked
about language structures (such as definition of truth) and you
indicated you understood that page, that Section.

That's not what I'm complaining here. My accusing you here has to do
with the fact that there and for more than 12+ months, _you understood_
_what I meant by language structure, structure theoretical truth_ as
evidenced from the link I've just given (and more if you requested);
and yet few posts ago you blamed me for your not understanding what
I meant there by "language structure".

Your flip-flopping on understanding my mentioned "language structure"
for your convenience is my accusing you there as not having a good-faith
in a technical argument.

>> You did claim to me that you are "a trained mathematician".
>>
>> I would expect straightforwardness and in-good-faith in every
>>
>> posters (including myself). But certainly from a trained mathematician!
>>
>
> I am being straightforward and arguing in good faith. You have no good reason to suppose otherwise. Your accusations to the contrary are unreasonable.
>

I've supplied evidences twice already to back up my accusation.

You did understand what I meant by language structure: for more than
one year you never complained you didn't understand what I had meant
by "language structure" _and_ you understood what I meant by it so as
your were able to have argued with me _from that understanding_ .

Now you've just conveniently denied that understanding of yours
(when I asked you about your failure to defend what you alluded to
before as an "incoherent structure").

[Rupert]

On Sunday, April 28, 2013 7:07:28 PM UTC+2, Newberry wrote:
> On Saturday, April 27, 2013 9:13:06 PM UTC-7, Rupert wrote:
> > I wasn't playing dumb. The context was not enough for me to know the answer to your question. Please do not be rude.
>
> > Somehow or other a human mathematician is able to engage in cognition about the natural numbers. There is no good reason to suppose that the brain processes that occur when this happens could not be adequately simulated by a computer.
>
>
>
> How can it be simulated if it cannot be captured by a finite number of axioms or axiom schemata?
>

The brain processes? Who says they can't be?

> > > Goedel's second theorem indicates that we cannot solve it.
>
> > That's false.
>
> Here is your contradiction. Franzen's proof purports to be absolute proof of ZFC consistency. Gödel's second theorem indicates that you can do so only in an infinite hierarchy of formal systems. Franzen somehow collapsed all the hierarchy into a few sentences.

Franzen's proof is a proof of the consistency of ZFC in ZFC*. There is no contradiction.

> > Let's say that ZFC is a theory in the first-order language of set theory, and that this language has a denumerable list of individual variables v_0, v_1, v_2, ...
>
> > Add a truth predicate True(x) to the language of ZFC. The intended interpretation is that True(x) will be true when x is an ordered pair whose first term is a Goedel code for a well-formed formula in the first-order language of set theory and whose second term is a finite sequence (x_0, x_1, ... x_n) such that all the free variables of the well-formed formula are among v_0, v_1, v_2, ... v_n, and the formula is true under any interpretation which assigns x_0 to v_0, x_1 to v_1, ... x_n to v_n.
>
>
> > Add to ZFC the obvious axioms governing the truth predicate which correspond to the Tarski clauses. Extend the axiom schema of Separation and the axiom schema of Replacement to include formulas which contain the truth predicate. Call the resulting theory ZFC*.
>
>
> > Franzen's proof can be formalized in ZFC*. This is not a counterexample to Goedel's second theorem. Goedel's second theorem applies to ZFC*. If ZFC* is consistent, then ZFC* cannot prove the consistency of ZFC*. But it can prove the consistency of ZFC.
>
>
> > So what's the problem?
>
>
>
> The problem is that this is NOT Franzen;s proof. (Firstly Franzen's proof is about consistency of ZFC not PA, but let's not dwell on this.)

I never mentioned PA.

> But mainly Franzen's proof is not relative to any other system whose consistency is unknown. Franzen's proof is absolute based on manifest truth.
>

What I gave you is a correct formalization of Franzen's proof.

> > > Well, the error is subtle. That is my whole point. I interpreted "truth preserving" as not losing any truth. (Maybe it is the long interpretation.) Anyway, the point is that there are some extra-axiomatic truths in classical logic i.e. tautologies, validities. Franzen did not examine if these are also manifestly true. In fact some of them are not and that is the error in his argument.
>
>
> > Right, so what is an example of a formula in the first-order language of set theory which is provable in the classical predicate calculus but is not manifestly true?
>
>
>
> Example 1:
>
> (P & ~P) -> Q
>
>
>
> Example 2:
>
> (x)((x+3 < x) -> (x = x+4))
>

This is not provable in the classical predicate calculus.

>
>
> If you want to argue from manifest truth then the paradox of material implication immediately becomes a suspect.

Wouldn't that just depend on what meaning you assigned to the connective -> ?

[Rupert]

On Sunday, April 28, 2013 7:50:53 PM UTC+2, Nam Nguyen wrote:
> On 27/04/2013 5:31 PM, Rupert wrote:
>
> > On Saturday, April 27, 2013 10:33:38 PM UTC+2, Nam Nguyen wrote:
>
>
>
> >>
>
> >> For the record, Rupert, your paragraph above has just proven I'm correct
>
> >>
>
> >> in complaining your argument style here is of the the nature of not-in
>
> >>
>
> >> good-faith (as well as being idiotic).
>
> >>
>
> >
>
> > Well, that's a very insulting thing to say, and it strikes me as being totally without rational foundation. I think you are being unreasonable.
>
>
>
> I don't accuse you without evidences (facts) to back it up.

Yes, you did.

Okay, well I apologize, I must have forgotten. Somehow or other I had got the impression that you meant something a bit different, especially when you talked about the "language structure of natural numbers" and its somehow not being fully determinate.

>
>
> That's not what I'm complaining here. My accusing you here has to do
>
> with the fact that there and for more than 12+ months, _you understood_
>
> _what I meant by language structure, structure theoretical truth_ as
>
> evidenced from the link I've just given (and more if you requested);
>
> and yet few posts ago you blamed me for your not understanding what
>
> I meant there by "language structure".
>

Look, I can't remember. If you really mean the same thing as Shoenfield, then that's fine, obviously. I have a bit of a suspicion I might be able to dig up some things you said in the past which seemed to indicate that you meant something different, but I'm not sure. If what you say is true, then it's just a failure of memory. It's not a good justification for insulting me and accusing me of not arguing in good faith.

>
>
> Your flip-flopping on understanding my mentioned "language structure"
>
> for your convenience is my accusing you there as not having a good-faith
>
> in a technical argument.
>

It's not for my convenience. Assuming that you really only ever meant the same thing as Shoenfield meant (which I'm not completely convinced of), then in that case it's just a case of my not remembering our previous conversation perfectly. It doesn't mean that I'm not arguing in good faith, and it's not a good reason to start abusing me.

>
>
> >> You did claim to me that you are "a trained mathematician".
>
> >>
>
> >> I would expect straightforwardness and in-good-faith in every
>
> >>
>
> >> posters (including myself). But certainly from a trained mathematician!
>
> >>
>
> >
>
> > I am being straightforward and arguing in good faith. You have no good reason to suppose otherwise. Your accusations to the contrary are unreasonable.
>
> >
>
>
>
> I've supplied evidences twice already to back up my accusation.
>

It's not good evidence. It doesn't show that I wasn't arguing in good faith. I was arguing in good faith, and you had no good reason to suppose otherwise. You ought to apologize for your intemperate accusation.

>
>
> You did understand what I meant by language structure: for more than
>
> one year you never complained you didn't understand what I had meant
>
> by "language structure" _and_ you understood what I meant by it so as
>
> your were able to have argued with me _from that understanding_ .
>

Perhaps. But I don't think I ever fully grasped what "structure-theoretic verification" was supposed to be exactly.

>
>
> Now you've just conveniently denied that understanding of yours
>
> (when I asked you about your failure to defend what you alluded to
>
> before as an "incoherent structure").
>

Well, you see, if by "structure" you really do mean the same thing that Shoenfield means, as you've just been claiming and indignantly taking me to task for not remembering, then the idea of an "incoherent structure" makes absolutely no sense whatsoever. The fact that you do seem to think that it somehow makes sense is very good reason for me to think that you don't really mean the same thing that Shoenfield means, and that I am entitled to ask you for clarification about what you do mean. So it was perfectly reasonable of me to do so, and all your ranting about how I'm not arguing in good faith is entirely unjustified and unreasonable, and you ought to apologize.

[Nam Nguyen]

On 29/04/2013 1:38 AM, Rupert wrote:
> On Sunday, April 28, 2013 7:50:53 PM UTC+2, Nam Nguyen wrote:
>> On 27/04/2013 5:31 PM, Rupert wrote:
>>
>>> I am being straightforward and arguing in good faith. You have no good reason to suppose otherwise. Your accusations to the contrary are unreasonable.
>>
>> I've supplied evidences twice already to back up my accusation.
>>
> It's not good evidence. It doesn't show that I wasn't arguing in good faith. I was arguing in good faith, and you had no good reason to suppose otherwise. You ought to apologize for your intemperate accusation.

I apologize then. I was frustrated from your seemingly position-changing
on your statements, however unintended it might have been (while the
debate/arguments seem to go no where). Please see more below.

>> You did understand what I meant by language structure: for more than
>>
>> one year you never complained you didn't understand what I had meant
>> by "language structure" _and_ you understood what I meant by it so as
>> your were able to have argued with me _from that understanding_ .
>>
>
> Perhaps. But I don't think I ever fully grasped what "structure-theoretic verification" was supposed to be exactly.

We could work and should work on that first then, before anything else,
since it's very important in the whole debate, your side or my side.

>>
>> Now you've just conveniently denied that understanding of yours
>>
>> (when I asked you about your failure to defend what you alluded to
>>
>> before as an "incoherent structure").
>>
>
> Well, you see, if by "structure" you really do mean the same thing that Shoenfield means, as you've just been claiming and indignantly taking me to task for not remembering, then the idea of an "incoherent structure" makes absolutely no sense whatsoever. The fact that you do seem to think that it somehow makes sense is very good reason for me to think that you don't really mean the same thing that Shoenfield means, and that I am entitled to ask you for clarification about what you do mean. So it was perfectly reasonable of me to do so, and all your ranting about how I'm not arguing in good faith is entirely unjustified and unreasonable, and you ought to apologize.

Well, you see, you forgot that the alluded phrase "incoherent
structure" was introduced by you (though indirectly), not by me.

I could pull the link if you request but basically in the argument
there _you_ had said if PA is inconsistent our understanding of the
naturals which _you_ also had agreed that the naturals are collectively
a language structure would be incoherent. That's why I asked you there
what you would mean by a structure being incoherent.

On my part, the terminology I had used was "incomplete" structure,
which I can define (as well as did).

In brief, "incoherent" is the term you introduced, not I.

This isn't the first time you forget things that would impact
the argument. Perhaps you could try to remember things better.

One of a way I could think in term of suggestion is to respond to-
the-point-the to the underlying key issues at foundational level,
before jumping too quick back to issues at higher level.

At this juncture would you be able to give me an acknowledgement
whether or not you'd understand on Def-1 and Def-2 I've mentioned
on the other thread:

https://groups.google.com/group/sci.logic/msg/6bd0f5919de69a33?hl=en

?

It has been a long argument, which is OK: it isn't an easy subject.

_We do need to make some progress one way or the other though_ imho.

--

[Newberry]

On Monday, April 29, 2013 12:29:33 AM UTC-7, Rupert wrote:
> On Sunday, April 28, 2013 7:07:28 PM UTC+2, Newberry wrote:
>
> > On Saturday, April 27, 2013 9:13:06 PM UTC-7, Rupert wrote:
>
> > > I wasn't playing dumb. The context was not enough for me to know the answer to your question. Please do not be rude.
>
> >
>
> > > Somehow or other a human mathematician is able to engage in cognition about the natural numbers. There is no good reason to suppose that the brain processes that occur when this happens could not be adequately simulated by a computer.
>
> >
>
> >
>
> >
>
> > How can it be simulated if it cannot be captured by a finite number of axioms or axiom schemata?
>
> >
>
>
>
> The brain processes? Who says they can't be?

Are you now saying that the brain functionality goes beyond the Turing machine?

And where does this natural numbers structure pre-exist? In the über-Turing brain, in Heaven, in an alternate universe. And how does it get transmitted to the über-Turing brain, by some kind of X-rays yet unknown to physics? How does all that work?

>
>
>
> > > > Goedel's second theorem indicates that we cannot solve it.
>
> >
>
> > > That's false.
>
> >
>
> > Here is your contradiction. Franzen's proof purports to be absolute proof of ZFC consistency. Gödel's second theorem indicates that you can do so only in an infinite hierarchy of formal systems. Franzen somehow collapsed all the hierarchy into a few sentences.
>
>
>
> Franzen's proof is a proof of the consistency of ZFC in ZFC*. There is no contradiction.

So you are saying that the following:
"The axioms of ZFC are manifestly true, the rules are truth preserving, therefore we derive nothing but truth, truth cannot be inconsistent therefore ZFC is consistent."
is the same thin as the formal consistency proof of ZFC in ZFC* [extended ZFC]? Where does the concept of manifest truth appear in the proof in ZFC*?

>
> > > Let's say that ZFC is a theory in the first-order language of set theory, and that this language has a denumerable list of individual variables v_0, v_1, v_2, ...
>
> >
>
> > > Add a truth predicate True(x) to the language of ZFC. The intended interpretation is that True(x) will be true when x is an ordered pair whose first term is a Goedel code for a well-formed formula in the first-order language of set theory and whose second term is a finite sequence (x_0, x_1, ... x_n) such that all the free variables of the well-formed formula are among v_0, v_1, v_2, ... v_n, and the formula is true under any interpretation which assigns x_0 to v_0, x_1 to v_1, ... x_n to v_n.
>
> >
>
> >
>
> > > Add to ZFC the obvious axioms governing the truth predicate which correspond to the Tarski clauses. Extend the axiom schema of Separation and the axiom schema of Replacement to include formulas which contain the truth predicate. Call the resulting theory ZFC*.
>
> >
>
> >
>
> > > Franzen's proof can be formalized in ZFC*. This is not a counterexample to Goedel's second theorem. Goedel's second theorem applies to ZFC*. If ZFC* is consistent, then ZFC* cannot prove the consistency of ZFC*. But it can prove the consistency of ZFC.
>
> >
>
> >
>
> > > So what's the problem?
>
> >
>
> >
>
> >
>
> > The problem is that this is NOT Franzen;s proof. (Firstly Franzen's proof is about consistency of ZFC not PA, but let's not dwell on this.)
>
>
>
> I never mentioned PA.
>
>
>
> > But mainly Franzen's proof is not relative to any other system whose consistency is unknown. Franzen's proof is absolute based on manifest truth.
>
> >
>
>
>
> What I gave you is a correct formalization of Franzen's proof.

So where is the manifest truth in the formalization?

>
>
>
> > > > Well, the error is subtle. That is my whole point. I interpreted "truth preserving" as not losing any truth. (Maybe it is the long interpretation.) Anyway, the point is that there are some extra-axiomatic truths in classical logic i.e. tautologies, validities. Franzen did not examine if these are also manifestly true. In fact some of them are not and that is the error in his argument.
>
> >
>
> >
>
> > > Right, so what is an example of a formula in the first-order language of set theory which is provable in the classical predicate calculus but is not manifestly true?
>
> >
>
> >
>
> >
>
> > Example 1:
>
> >
>
> > (P & ~P) -> Q
>
> >
>
> >
>
> >
>
> > Example 2:
>
> >
>
> > (x)((x+3 < x) -> (x = x+4))
>
> >
>
>
>
> This is not provable in the classical predicate calculus.

Here is what you said in another thread:
"You mean when the hypothesis of the conditional is false? That's incorrect. If
the hypothesis of the conditional is false, then the formula is vacuously true."

>
>
>
> >
>
> >
>
> > If you want to argue from manifest truth then the paradox of material implication immediately becomes a suspect.
>
>
>
> Wouldn't that just depend on what meaning you assigned to the connective -> ?

I tend to think not. But in any case if you interpret A -> B as ~P v Q and 'v' as the Boolean '+' then you get a result that is counter-intuitive [(P & ~P) -> Q] i.e. not manifestly true.

[Rupert]

On Tuesday, April 30, 2013 4:00:47 AM UTC+2, Newberry wrote:
> On Monday, April 29, 2013 12:29:33 AM UTC-7, Rupert wrote:
>
> > On Sunday, April 28, 2013 7:07:28 PM UTC+2, Newberry wrote:
>
> >
>
> > > On Saturday, April 27, 2013 9:13:06 PM UTC-7, Rupert wrote:
>
> >
>
> > > > I wasn't playing dumb. The context was not enough for me to know the answer to your question. Please do not be rude.
>
> >
>
> > >
>
> >
>
> > > > Somehow or other a human mathematician is able to engage in cognition about the natural numbers. There is no good reason to suppose that the brain processes that occur when this happens could not be adequately simulated by a computer.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > How can it be simulated if it cannot be captured by a finite number of axioms or axiom schemata?
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > The brain processes? Who says they can't be?
>
> Are you now saying that the brain functionality goes beyond the Turing machine?
>

No.

>
>
> And where does this natural numbers structure pre-exist? In the über-Turing brain, in Heaven, in an alternate universe. And how does it get transmitted to the über-Turing brain, by some kind of X-rays yet unknown to physics? How does all that work?
>

You should take a look at Geoffrey Hellman's "Mathematics Without Numbers".

> >
>
> >
>
> >
>
> > > > > Goedel's second theorem indicates that we cannot solve it.
>
> >
>
> > >
>
> >
>
> > > > That's false.
>
> >
>
> > >
>
> >
>
> > > Here is your contradiction. Franzen's proof purports to be absolute proof of ZFC consistency. Gödel's second theorem indicates that you can do so only in an infinite hierarchy of formal systems. Franzen somehow collapsed all the hierarchy into a few sentences.
>
> >
>
> >
>
> >
>
> > Franzen's proof is a proof of the consistency of ZFC in ZFC*. There is no contradiction.
>
> So you are saying that the following:
>
> "The axioms of ZFC are manifestly true, the rules are truth preserving, therefore we derive nothing but truth, truth cannot be inconsistent therefore ZFC is consistent."
>
> is the same thin as the formal consistency proof of ZFC in ZFC* [extended ZFC]? Where does the concept of manifest truth appear in the proof in ZFC*?
>

In the truth predicate.

> >
>
> > > > Let's say that ZFC is a theory in the first-order language of set theory, and that this language has a denumerable list of individual variables v_0, v_1, v_2, ...
>
> >
>
> > >
>
> >
>
> > > > Add a truth predicate True(x) to the language of ZFC. The intended interpretation is that True(x) will be true when x is an ordered pair whose first term is a Goedel code for a well-formed formula in the first-order language of set theory and whose second term is a finite sequence (x_0, x_1, ... x_n) such that all the free variables of the well-formed formula are among v_0, v_1, v_2, ... v_n, and the formula is true under any interpretation which assigns x_0 to v_0, x_1 to v_1, ... x_n to v_n.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > Add to ZFC the obvious axioms governing the truth predicate which correspond to the Tarski clauses. Extend the axiom schema of Separation and the axiom schema of Replacement to include formulas which contain the truth predicate. Call the resulting theory ZFC*.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > Franzen's proof can be formalized in ZFC*. This is not a counterexample to Goedel's second theorem. Goedel's second theorem applies to ZFC*. If ZFC* is consistent, then ZFC* cannot prove the consistency of ZFC*. But it can prove the consistency of ZFC.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > So what's the problem?
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > The problem is that this is NOT Franzen;s proof. (Firstly Franzen's proof is about consistency of ZFC not PA, but let's not dwell on this.)
>
> >
>
> >
>
> >
>
> > I never mentioned PA.
>
> >
>
> >
>
> >
>
> > > But mainly Franzen's proof is not relative to any other system whose consistency is unknown. Franzen's proof is absolute based on manifest truth.
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > What I gave you is a correct formalization of Franzen's proof.
>
> So where is the manifest truth in the formalization?
>

In the truth predicate.

> >
>
> >
>
> >
>
> > > > > Well, the error is subtle. That is my whole point. I interpreted "truth preserving" as not losing any truth. (Maybe it is the long interpretation.) Anyway, the point is that there are some extra-axiomatic truths in classical logic i.e. tautologies, validities. Franzen did not examine if these are also manifestly true. In fact some of them are not and that is the error in his argument.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > Right, so what is an example of a formula in the first-order language of set theory which is provable in the classical predicate calculus but is not manifestly true?
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Example 1:
>
> >
>
> > >
>
> >
>
> > > (P & ~P) -> Q
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Example 2:
>
> >
>
> > >
>
> >
>
> > > (x)((x+3 < x) -> (x = x+4))
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > This is not provable in the classical predicate calculus.
>
> Here is what you said in another thread:
>
> "You mean when the hypothesis of the conditional is false? That's incorrect. If
>
> the hypothesis of the conditional is false, then the formula is vacuously true."
>

Yes, but in this case the falsity of the hypothesis is not provable in the classical predicate calculus. You need some nonlogical axioms of number theory to get it.

>
>
> >
>
> >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > If you want to argue from manifest truth then the paradox of material implication immediately becomes a suspect.
>
> >
>
> >
>
> >
>
> > Wouldn't that just depend on what meaning you assigned to the connective -> ?
>
> I tend to think not. But in any case if you interpret A -> B as ~P v Q and 'v' as the Boolean '+' then you get a result that is counter-intuitive [(P & ~P) -> Q] i.e. not manifestly true.

Well, surely it is manifestly true on the proposed meaning for -> .

[Rupert]

On Monday, April 29, 2013 5:37:35 PM UTC+2, Nam Nguyen wrote:
> On 29/04/2013 1:38 AM, Rupert wrote:
>
> > On Sunday, April 28, 2013 7:50:53 PM UTC+2, Nam Nguyen wrote:
>
> >> On 27/04/2013 5:31 PM, Rupert wrote:
>
> >>
>
> >>> I am being straightforward and arguing in good faith. You have no good reason to suppose otherwise. Your accusations to the contrary are unreasonable.
>
> >>
>
> >> I've supplied evidences twice already to back up my accusation.
>
> >>
>
> > It's not good evidence. It doesn't show that I wasn't arguing in good faith. I was arguing in good faith, and you had no good reason to suppose otherwise. You ought to apologize for your intemperate accusation.
>
>
>
> I apologize then. I was frustrated from your seemingly position-changing
>
> on your statements, however unintended it might have been (while the
>
> debate/arguments seem to go no where). Please see more below.
>
>
>
> >> You did understand what I meant by language structure: for more than
>
> >>
>
> >> one year you never complained you didn't understand what I had meant
>
> >> by "language structure" _and_ you understood what I meant by it so as
>
> >> your were able to have argued with me _from that understanding_ .
>
> >>
>
> >
>
> > Perhaps. But I don't think I ever fully grasped what "structure-theoretic verification" was supposed to be exactly.
>
>
>
> We could work and should work on that first then, before anything else,
>
> since it's very important in the whole debate, your side or my side.
>
>
>
> >>
>
> >> Now you've just conveniently denied that understanding of yours
>
> >>
>
> >> (when I asked you about your failure to defend what you alluded to
>
> >>
>
> >> before as an "incoherent structure").
>
> >>
>
> >
>
> > Well, you see, if by "structure" you really do mean the same thing that Shoenfield means, as you've just been claiming and indignantly taking me to task for not remembering, then the idea of an "incoherent structure" makes absolutely no sense whatsoever. The fact that you do seem to think that it somehow makes sense is very good reason for me to think that you don't really mean the same thing that Shoenfield means, and that I am entitled to ask you for clarification about what you do mean. So it was perfectly reasonable of me to do so, and all your ranting about how I'm not arguing in good faith is entirely unjustified and unreasonable, and you ought to apologize.
>
>
>
> Well, you see, you forgot that the alluded phrase "incoherent
>
> structure" was introduced by you (though indirectly), not by me.
>

I find that very difficult to believe.

>
>
> I could pull the link if you request but basically in the argument
>
> there _you_ had said if PA is inconsistent our understanding of the
>
> naturals which _you_ also had agreed that the naturals are collectively
>
> a language structure would be incoherent.

If PA is inconsistent, then we don't have a coherent conception of the natural numbers. That's not the same as saying that they form an "incoherent structure". And when I grant that they do form a structure, that's only assuming certain assumptions, which are probably strong enough to prove the consistency of PA.

> That's why I asked you there
>
> what you would mean by a structure being incoherent.
>

I don't mean anything by it. I never used that phrase. It doesn't mean anything.

>
>
> On my part, the terminology I had used was "incomplete" structure,
>
> which I can define (as well as did).
>

I can't remember what the definition was.

>
>
> In brief, "incoherent" is the term you introduced, not I.
>

I didn't introduce the notion of an "incoherent structure".

>
>
> This isn't the first time you forget things that would impact
>
> the argument. Perhaps you could try to remember things better.
>

Perhaps you could stop attributing things to me that I never said.

>
>
> One of a way I could think in term of suggestion is to respond to-
>
> the-point-the to the underlying key issues at foundational level,
>
> before jumping too quick back to issues at higher level.
>
>
>
> At this juncture would you be able to give me an acknowledgement
>
> whether or not you'd understand on Def-1 and Def-2 I've mentioned
>
> on the other thread:
>
>
>
> https://groups.google.com/group/sci.logic/msg/6bd0f5919de69a33?hl=en
>
>
>
> ?
>

No, I don't understand those definitions.

[Nam Nguyen]

On 01/05/2013 11:24 AM, Rupert wrote:
> On Monday, April 29, 2013 5:37:35 PM UTC+2, Nam Nguyen wrote:
>>
>>
>> I could pull the link if you request but basically in the argument
>> there _you_ had said if PA is inconsistent our understanding of the
>> naturals which _you_ also had agreed that the naturals are collectively
>> a language structure would be incoherent.
>
> If PA is inconsistent, then we don't have a coherent conception of the natural numbers.

Technically translate your meta statement above, we'd have:

(*) (PA |- (Ax[x=x] /\ ~Ax[x=x])) => "The concept of the natural numbers
is incoherent".

Obviously if you can _not_ prove (*) you wouldn't know what you're
talking about.

Can you prove (*)?

[Rupert]

On Thursday, May 2, 2013 4:51:45 AM UTC+2, Nam Nguyen wrote:
> On 01/05/2013 11:24 AM, Rupert wrote:
>
> > On Monday, April 29, 2013 5:37:35 PM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>
>
> >> I could pull the link if you request but basically in the argument
>
> >> there _you_ had said if PA is inconsistent our understanding of the
>
> >> naturals which _you_ also had agreed that the naturals are collectively
>
> >> a language structure would be incoherent.
>
> >
>
> > If PA is inconsistent, then we don't have a coherent conception of the natural numbers.
>
>
>
> Technically translate your meta statement above, we'd have:
>
>
>
> (*) (PA |- (Ax[x=x] /\ ~Ax[x=x])) => "The concept of the natural numbers
>
> is incoherent".
>
>
>
> Obviously if you can _not_ prove (*) you wouldn't know what you're
>
> talking about.
>
>
>
> Can you prove (*)?
>

It's not meant to be an assertion that is capable of mathematical proof. It's a comment about what is essential to our conception of the natural numbers. It is essential to our conception of the natural numbers that they satisfy the axioms of PA. If those axioms are inconsistent then the conception is not coherent. Pretty simple really.

[Nam Nguyen]

On 01/05/2013 10:24 PM, Rupert wrote:
> On Thursday, May 2, 2013 4:51:45 AM UTC+2, Nam Nguyen wrote:
>> On 01/05/2013 11:24 AM, Rupert wrote:
>>
>>> On Monday, April 29, 2013 5:37:35 PM UTC+2, Nam Nguyen wrote:
>>
>>>>
>>
>>>>
>>
>>>> I could pull the link if you request but basically in the argument
>>
>>>> there _you_ had said if PA is inconsistent our understanding of the
>>
>>>> naturals which _you_ also had agreed that the naturals are collectively
>>
>>>> a language structure would be incoherent.
>>
>>>
>>
>>> If PA is inconsistent, then we don't have a coherent conception of the natural numbers.
>>
>>
>>
>> Technically translate your meta statement above, we'd have:
>>
>>
>> (*) (PA |- (Ax[x=x] /\ ~Ax[x=x])) => "The concept of the natural numbers
>>
>> is incoherent".
>>
>>
>>
>> Obviously if you can _not_ prove (*) you wouldn't know what you're
>>
>> talking about.
>>
>>
>>
>> Can you prove (*)?
>>
>
> It's not meant to be an assertion that is capable of mathematical proof.

Right. It's not meant to be a proof of a FOL language formula.
But there's such a _meta-mathematical proof_ for a meta-mathematical
statement too.

And so far you've failed to provide such a meta-mathematical proof.

> It's a comment about what is essential to our conception of the natural numbers. It is essential to our conception of the natural numbers that they satisfy the axioms of PA.

"Essential to our conception of the natural numbers" is a meaningless
statement of reasoning, when you've so far failed to know the truth
value of a statement, e.g., cGC.

> If those axioms are inconsistent then the conception is not coherent. Pretty simple really.

It's a pretty simple wishful thinking. In fact, it's is an incorrect
statement.

It's a very _basic knowledge of FOL_ that _if the axioms are_
_inconsistent_ then _they're inconsistent because of themselves_ .
Period.

[Rupert]

On Thursday, May 2, 2013 7:49:57 AM UTC+2, Nam Nguyen wrote:
> On 01/05/2013 10:24 PM, Rupert wrote:
>
> > On Thursday, May 2, 2013 4:51:45 AM UTC+2, Nam Nguyen wrote:
>
> >> On 01/05/2013 11:24 AM, Rupert wrote:
>
> > It's not meant to be an assertion that is capable of mathematical proof.
>
>
>
> Right. It's not meant to be a proof of a FOL language formula.
>
> But there's such a _meta-mathematical proof_ for a meta-mathematical
>
> statement too.
>

It's not really a meta-mathematical statement. It's a remark about the content of our conception of the natural numbers.

>
>
> And so far you've failed to provide such a meta-mathematical proof.
>

Let me ask you something. Do you believe that it is part of our conception of the natural numbers that they satisfy the PA axioms? Or not?

>
>
> > It's a comment about what is essential to our conception of the natural numbers. It is essential to our conception of the natural numbers that they satisfy the axioms of PA.
>
>
>
> "Essential to our conception of the natural numbers" is a meaningless
>
> statement of reasoning, when you've so far failed to know the truth
>
> value of a statement, e.g., cGC.
>

It's not meaningless. It has a perfectly good meaning. The fact that I don't know the truth-value of cGC has nothing to do with it.

>
>
> > If those axioms are inconsistent then the conception is not coherent. Pretty simple really.
>
>
>
> It's a pretty simple wishful thinking. In fact, it's is an incorrect
>
> statement.
>

Suppose that Q were discovered to be inconsistent. Would you say then that we have a coherent conception of the natural numbers?

>
>
> It's a very _basic knowledge of FOL_ that _if the axioms are_
>
> _inconsistent_ then _they're inconsistent because of themselves_ .
>
> Period.
>

Irrelevant.

[Nam Nguyen]

Apparently your argument throughout seems to be based on the notion
that certain statements written in the language of arithmetic be
_truly reflecting some arithmetic facts_ simply because they would
_mean_ the fact.

Such notion is simply an illusion: there might be arithmetic fact
that no formula written in the language of arithmetic could express,
and as such the truth of the fact can't be expressed, lest alone
be proven, be known!

For instance, it's still a possibility that there are finitely many
counter examples of Goldbach Conjecture.

But, to begin with, how would you really express:

"there are finitely many counter examples of Goldbach Conjecture"

_as a formula_ written in the language of arithmetic L(PA)?

[Nam Nguyen]

On 02/05/2013 12:38 AM, Rupert wrote:
> On Thursday, May 2, 2013 7:49:57 AM UTC+2, Nam Nguyen wrote:
>> On 01/05/2013 10:24 PM, Rupert wrote:
>>
>>> On Thursday, May 2, 2013 4:51:45 AM UTC+2, Nam Nguyen wrote:
>>
>>>> On 01/05/2013 11:24 AM, Rupert wrote:
>>
>>> It's not meant to be an assertion that is capable of mathematical proof.
>>
>>
>> Right. It's not meant to be a proof of a FOL language formula.
>>
>> But there's such a _meta-mathematical proof_ for a meta-mathematical
>> statement too.
>>
>
> It's not really a meta-mathematical statement. It's a remark about the content of our conception of the natural numbers.

I don't know what "not really" logically means: it's either a
meta-mathematical statement, assertion, or it is not. And as far
as one can tell, it is meta-mathematical assertion, without proof.

>
>> And so far you've failed to provide such a meta-mathematical proof.
>>
> Let me ask you something. Do you believe that it is part of our conception of the natural numbers that they satisfy the PA axioms? Or not?

Since when "belief" is a part of logical, mathematical reasoning?
Today I believe so. Why can't I _change_ my belief tomorrow?

>
>>
>>> It's a comment about what is essential to our conception of the natural numbers. It is essential to our conception of the natural numbers that they satisfy the axioms of PA.
>>
>> "Essential to our conception of the natural numbers" is a meaningless
>>
>> statement of reasoning, when you've so far failed to know the truth
>>
>> value of a statement, e.g., cGC.
>>
>
> It's not meaningless. It has a perfectly good meaning. The fact that I don't know the truth-value of cGC has nothing to do with it.

It does: it means the consistency, inconsistency (whatever the case be)
has no _logical connection_ with the concept of the natural numbers.

>>
>>> If those axioms are inconsistent then the conception is not coherent. Pretty simple really.
>>
>> It's a pretty simple wishful thinking. In fact, it's is an incorrect
>> statement.
>>
>
> Suppose that Q were discovered to be inconsistent. Would you say then that we have a coherent conception of the natural numbers?

But what is the "natural numbers" to begin with?

Are they the "natural modulo-4 numbers" where 2+2=0? I'd guess not.

How about they're the kind of "natural numbers" where ~cGC is true?

How exactly would you _DEFINE_ the "natural numbers" _when_
_your DEFINITION does NOT ENTAIL the truth of 1 finite formula_ ?

In any rate your question is meaningless since the definition of formal
system provability is _independent_ from the definition of structure
theoretical truth, and vice versa.

>>
>> It's a very _basic knowledge of FOL_ that _if the axioms are_
>> _inconsistent_ then _they're inconsistent because of themselves_ .
>>
>> Period.
>>
> Irrelevant.
>

You're wrong: it's relevant.

[Rupert]

On Thursday, May 2, 2013 9:05:56 AM UTC+2, Nam Nguyen wrote:
> On 02/05/2013 12:38 AM, Rupert wrote:
>
> > On Thursday, May 2, 2013 7:49:57 AM UTC+2, Nam Nguyen wrote:
>
> >> On 01/05/2013 10:24 PM, Rupert wrote:
>
> >>
>
> >>> On Thursday, May 2, 2013 4:51:45 AM UTC+2, Nam Nguyen wrote:
>
> >>
>
> >>>> On 01/05/2013 11:24 AM, Rupert wrote:
>
> >>
>
> >>> It's not meant to be an assertion that is capable of mathematical proof.
>
> >>
>
> >>
>
> >> Right. It's not meant to be a proof of a FOL language formula.
>
> >>
>
> >> But there's such a _meta-mathematical proof_ for a meta-mathematical
>
> >> statement too.
>
> >>
>
> >
>
> > It's not really a meta-mathematical statement. It's a remark about the content of our conception of the natural numbers.
>
>
>
> I don't know what "not really" logically means: it's either a
>
> meta-mathematical statement, assertion, or it is not.

Well, I told you it isn't, didn't I?

> And as far
>
> as one can tell, it is meta-mathematical assertion, without proof.
>

Why do you think it's a metamathematical assertion?

> >
>
> >> And so far you've failed to provide such a meta-mathematical proof.
>
> >>
>
> > Let me ask you something. Do you believe that it is part of our conception of the natural numbers that they satisfy the PA axioms? Or not?
>
>
>
> Since when "belief" is a part of logical, mathematical reasoning?
>
> Today I believe so. Why can't I _change_ my belief tomorrow?
>

Today you believe that it is a part of our conception of the natural numbers that they satisfy the PA axioms. So presumably you should grant that if the PA axioms were shown to be inconsistent then it would follow that that conception is incoherent. No?

> >
>
> >>
>
> >>> It's a comment about what is essential to our conception of the natural numbers. It is essential to our conception of the natural numbers that they satisfy the axioms of PA.
>
> >>
>
> >> "Essential to our conception of the natural numbers" is a meaningless
>
> >>
>
> >> statement of reasoning, when you've so far failed to know the truth
>
> >>
>
> >> value of a statement, e.g., cGC.
>
> >>
>
> >
>
> > It's not meaningless. It has a perfectly good meaning. The fact that I don't know the truth-value of cGC has nothing to do with it.
>
>
>
> It does: it means the consistency, inconsistency (whatever the case be)
>
> has no _logical connection_ with the concept of the natural numbers.
>

No. It doesn't mean that.

>
>
> >>
>
> >>> If those axioms are inconsistent then the conception is not coherent. Pretty simple really.
>
> >>
>
> >> It's a pretty simple wishful thinking. In fact, it's is an incorrect
>
> >> statement.
>
> >>
>
> >
>
> > Suppose that Q were discovered to be inconsistent. Would you say then that we have a coherent conception of the natural numbers?
>
>
>
> But what is the "natural numbers" to begin with?
>

Well, I assume you've learnt how to count? You are familiar with the counting numbers 1, 2, 3, 4 ... ? And you have also learnt about 0? No?

>
>
> Are they the "natural modulo-4 numbers" where 2+2=0? I'd guess not.
>

Indeed not.

>
>
> How about they're the kind of "natural numbers" where ~cGC is true?
>

Do you want to claim that there is such a "kind of natural numbers"? That claim requires some argument.

>
>
> How exactly would you _DEFINE_ the "natural numbers" _when_
>
> _your DEFINITION does NOT ENTAIL the truth of 1 finite formula_ ?
>

Giving a mathematical definition of the natural numbers would be a different matter, and you would not be in a position to know whether my definition does or does not entail the truth of falsity of cGC.

I am happy to give you a definition of the natural numbers working in the theory ZFC if it is of any interest to you.

>
>
> In any rate your question is meaningless since the definition of formal
>
> system provability is _independent_ from the definition of structure
>
> theoretical truth, and vice versa.
>

It's not meaningless. Just "yes" or "no" would be great, thanks.

>
>
> >>
>
> >> It's a very _basic knowledge of FOL_ that _if the axioms are_
>
> >> _inconsistent_ then _they're inconsistent because of themselves_ .
>
> >>
>
> >> Period.
>
> >>
>
> > Irrelevant.
>
> >
>
> You're wrong: it's relevant.
>

Why?

[Rupert]

(En)(Am)(m>n -> ((Ek)m=2k+1 v (Ep)(Eq)(p>1&q>1&p+q=m&(Ar)(((Es)rs=p)->(r=1 v r=p))&(Ar)(((Es)rs=q)->(r-1 v r=q)))))

[Newberry]

On Wednesday, May 1, 2013 10:17:41 AM UTC-7, Rupert wrote:
> On Tuesday, April 30, 2013 4:00:47 AM UTC+2, Newberry wrote:
>
> > On Monday, April 29, 2013 12:29:33 AM UTC-7, Rupert wrote:
>
> >
>
> > > On Sunday, April 28, 2013 7:07:28 PM UTC+2, Newberry wrote:
>
> >
>
> > >
>
> >
>
> > > > On Saturday, April 27, 2013 9:13:06 PM UTC-7, Rupert wrote:
>
> >
>
> > >
>
> >
>
> > > > > I wasn't playing dumb. The context was not enough for me to know the answer to your question. Please do not be rude.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > Somehow or other a human mathematician is able to engage in cognition about the natural numbers. There is no good reason to suppose that the brain processes that occur when this happens could not be adequately simulated by a computer.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > How can it be simulated if it cannot be captured by a finite number of axioms or axiom schemata?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > The brain processes? Who says they can't be?
>
> >
>
> > Are you now saying that the brain functionality goes beyond the Turing machine?
>
> >
>
>
>
> No.

You lost me.

>
>
>
> >
>
> >
>
> > And where does this natural numbers structure pre-exist? In the über-Turing brain, in Heaven, in an alternate universe. And how does it get transmitted to the über-Turing brain, by some kind of X-rays yet unknown to physics? How does all that work?
>
> >
>
>
>
> You should take a look at Geoffrey Hellman's "Mathematics Without Numbers".

I would be interested what you had to say about this, not some third party.

>
>
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > > > Goedel's second theorem indicates that we cannot solve it.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > That's false.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > Here is your contradiction. Franzen's proof purports to be absolute proof of ZFC consistency. Gödel's second theorem indicates that you can do so only in an infinite hierarchy of formal systems. Franzen somehow collapsed all the hierarchy into a few sentences.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Franzen's proof is a proof of the consistency of ZFC in ZFC*. There is no contradiction.
>
> >
>
> > So you are saying that the following:
>
> >
>
> > "The axioms of ZFC are manifestly true, the rules are truth preserving, therefore we derive nothing but truth, truth cannot be inconsistent therefore ZFC is consistent."
>
> >
>
> > is the same thin as the formal consistency proof of ZFC in ZFC* [extended ZFC]? Where does the concept of manifest truth appear in the proof in ZFC*?
>
> >
>
>
>
> In the truth predicate.

How do you express "Axiom A1 is manifestly true" in ZFC*?

>
>
>
> > >
>
> >
>
> > > > > Let's say that ZFC is a theory in the first-order language of set theory, and that this language has a denumerable list of individual variables v_0, v_1, v_2, ...
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > Add a truth predicate True(x) to the language of ZFC. The intended interpretation is that True(x) will be true when x is an ordered pair whose first term is a Goedel code for a well-formed formula in the first-order language of set theory and whose second term is a finite sequence (x_0, x_1, ... x_n) such that all the free variables of the well-formed formula are among v_0, v_1, v_2, ... v_n, and the formula is true under any interpretation which assigns x_0 to v_0, x_1 to v_1, ... x_n to v_n.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > Add to ZFC the obvious axioms governing the truth predicate which correspond to the Tarski clauses. Extend the axiom schema of Separation and the axiom schema of Replacement to include formulas which contain the truth predicate. Call the resulting theory ZFC*.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > Franzen's proof can be formalized in ZFC*. This is not a counterexample to Goedel's second theorem. Goedel's second theorem applies to ZFC*. If ZFC* is consistent, then ZFC* cannot prove the consistency of ZFC*. But it can prove the consistency of ZFC.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > So what's the problem?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > The problem is that this is NOT Franzen;s proof. (Firstly Franzen's proof is about consistency of ZFC not PA, but let's not dwell on this.)
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > I never mentioned PA.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > But mainly Franzen's proof is not relative to any other system whose consistency is unknown. Franzen's proof is absolute based on manifest truth.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > What I gave you is a correct formalization of Franzen's proof.
>
> >
>
> > So where is the manifest truth in the formalization?
>
> >
>
>
>
> In the truth predicate.
>
>
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > > > > Well, the error is subtle. That is my whole point. I interpreted "truth preserving" as not losing any truth. (Maybe it is the long interpretation.) Anyway, the point is that there are some extra-axiomatic truths in classical logic i.e. tautologies, validities. Franzen did not examine if these are also manifestly true. In fact some of them are not and that is the error in his argument.
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > Right, so what is an example of a formula in the first-order language of set theory which is provable in the classical predicate calculus but is not manifestly true?
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > Example 1:
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > (P & ~P) -> Q
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > Example 2:
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > (x)((x+3 < x) -> (x = x+4))
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > This is not provable in the classical predicate calculus.
>
> >
>
> > Here is what you said in another thread:
>
> >
>
> > "You mean when the hypothesis of the conditional is false? That's incorrect. If
>
> >
>
> > the hypothesis of the conditional is false, then the formula is vacuously true."
>
> >
>
>
>
> Yes, but in this case the falsity of the hypothesis is not provable in the classical predicate calculus. You need some nonlogical axioms of number theory to get it.

(x)((x#x) -> (x = x+4))
Not sure what you are trying to accomplish but this nitpicking. The point is that classical predicate calculus induces this kind of spurious "truths".

>
>
>
> >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > If you want to argue from manifest truth then the paradox of material implication immediately becomes a suspect.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Wouldn't that just depend on what meaning you assigned to the connective -> ?
>
> >
>
> > I tend to think not. But in any case if you interpret A -> B as ~P v Q and 'v' as the Boolean '+' then you get a result that is counter-intuitive [(P & ~P) -> Q] i.e. not manifestly true.
>
>
>
> Well, surely it is manifestly true on the proposed meaning for -> .

You are confusing things quite badly here. If you define Boolean +, ., ~ the usual way then it is true that
~(A & ~A) v B (1)
is always one. That does NOT mean that
"If it rains and it does not rain then it is sunny"
is true. You cannot define truth arbitrarily.

BTW, that (1) is always 1 is true is analytic, and I do not think that is what Franzen means by manifest truth.

Truth means correspondence with reality. So if there is no king of Switzerland on the list of the wise people then you cannot say that "all the kings of Switzerland have been wise" is true. The same applies to observations on N. But if you do do such silly things then you *will* have problems with pi_1 sentences.

[Rupert]

On Thursday, May 2, 2013 4:01:10 PM UTC+2, Newberry wrote:
> On Wednesday, May 1, 2013 10:17:41 AM UTC-7, Rupert wrote:
>
> > On Tuesday, April 30, 2013 4:00:47 AM UTC+2, Newberry wrote:
>
> >
>
> > > On Monday, April 29, 2013 12:29:33 AM UTC-7, Rupert wrote:
>

> > No.
>
> You lost me.
>

I never said anything which in any way implied that the brain processes can't be captured by a Turing machine. I suggested that maybe they can be captured by a finite number of axioms or axiom schemata. If anything this would help the idea that they could be captured by a Turing machine.

> > You should take a look at Geoffrey Hellman's "Mathematics Without Numbers".
>
> I would be interested what you had to say about this, not some third party.
>

Well, the basic idea is that mathematical assertions can be re-interpreted in a language with modal operators and second-order quantifiers. The question of how we come to have knowledge of the mathematical assertions is an interesting one, but you could maybe construct some kind of story about it based on our experience with actual finite structures and our intuitions about logical possibilities.

It's a bit difficult to sum up all in one go. I would suggest that you read the book and then get back to me.

> > In the truth predicate.
>
> How do you express "Axiom A1 is manifestly true" in ZFC*?
>

Apply the truth predicate to the Goedel code for axiom A1.

> > Yes, but in this case the falsity of the hypothesis is not provable in the classical predicate calculus. You need some nonlogical axioms of number theory to get it.
>
> (x)((x#x) -> (x = x+4))
>
> Not sure what you are trying to accomplish but this nitpicking. The point is that classical predicate calculus induces this kind of spurious "truths".
>

Right, so why is that not manifestly true?

> > Well, surely it is manifestly true on the proposed meaning for -> .
>
> You are confusing things quite badly here. If you define Boolean +, ., ~ the usual way then it is true that
>
> ~(A & ~A) v B (1)
>
> is always one. That does NOT mean that
>
> "If it rains and it does not rain then it is sunny"
>
> is true. You cannot define truth arbitrarily.
>

Why isn't it true?

>
>
> BTW, that (1) is always 1 is true is analytic, and I do not think that is what Franzen means by manifest truth.
>
>
>
> Truth means correspondence with reality. So if there is no king of Switzerland on the list of the wise people then you cannot say that "all the kings of Switzerland have been wise" is true.

Why not?

> The same applies to observations on N. But if you do do such silly things then you *will* have problems with pi_1 sentences.

You still haven't given me an example of a pi_1 sentence which is true on the standard semantics but not true on your semantics.

[Nam Nguyen]

Wrong.

"There are _zero_ counter examples" should still be _a finite case_
but which _your formula does NOT express_ !

[Rupert]

On Friday, May 3, 2013 2:48:54 AM UTC+2, Nam Nguyen wrote:
> On 02/05/2013 2:40 AM, Rupert wrote:
>
> > On Thursday, May 2, 2013 8:44:29 AM UTC+2, Nam Nguyen wrote:
>
>
>
> >> Apparently your argument throughout seems to be based on the notion
>
> >> that certain statements written in the language of arithmetic be
>
> >> _truly reflecting some arithmetic facts_ simply because they would
>
> >> _mean_ the fact.
>
> >>
>
> >> Such notion is simply an illusion: there might be arithmetic fact
>
> >> that no formula written in the language of arithmetic could express,
>
> >> and as such the truth of the fact can't be expressed, lest alone
>
> >> be proven, be known!
>
> >>
>
> >> For instance, it's still a possibility that there are finitely many
>
> >> counter examples of Goldbach Conjecture.
>
> >>
>
> >> But, to begin with, how would you really express:
>
> >>
>
> >> "there are finitely many counter examples of Goldbach Conjecture"
>
> >>
>
> >> _as a formula_ written in the language of arithmetic L(PA)?
>
> >
>
> > (En)(Am)(m>n -> ((Ek)m=2k+1 v (Ep)(Eq)(p>1&q>1&p+q=m&(Ar)(((Es)rs=p)->(r=1 v r=p))&(Ar)(((Es)rs=q)->(r-1 v r=q)))))
>
>
>
> Wrong.
>
>
>
> "There are _zero_ counter examples" should still be _a finite case_
>

That's covered by what I wrote.

> but which _your formula does NOT express_ !
>

Why not?

>

[Newberry]

On Friday, April 26, 2013 7:24:01 PM UTC-7, fom wrote:
> On 4/24/2013 8:28 PM, Newberry wrote:
>
> > So don't
>
> > tell me that it is all based on manifest truth. In fact I have shown
>
> > in another thread
>
> > https://groups.google.com/forum/?hl=en&fromgroups#!topic/sci.logic/lDJcgOg4vco
>
> > that the proof that the truths of first order arithmetic are not
>
> > recursively enumerable is NOT likely to hold if we use Strawson-like
>
> > semantics.
>
> >
>
>
>
> What exactly do you mean by "Strawson-like"
>
> semantics. Did Strawson ever produce a semantic
>
> theory?

I did produce Strawson-like semantic theory.

[Newberry]

On Thursday, May 2, 2013 10:08:56 AM UTC-7, Rupert wrote:
> On Thursday, May 2, 2013 4:01:10 PM UTC+2, Newberry wrote:
>
> > On Wednesday, May 1, 2013 10:17:41 AM UTC-7, Rupert wrote:
>
> >
>
> > > On Tuesday, April 30, 2013 4:00:47 AM UTC+2, Newberry wrote:
>
> >
>
> > >
>
> >
>
> > > > On Monday, April 29, 2013 12:29:33 AM UTC-7, Rupert wrote:
>
> >
>
> > > No.
>
> >
>
> > You lost me.
>
> >
>
>
>
> I never said anything which in any way implied that the brain processes can't be captured by a Turing machine. I suggested that maybe they can be captured by a finite number of axioms or axiom schemata. If anything this would help the idea that they could be captured by a Turing machine.

Then I do not know how the brain can fathom the structure of natural numbers N, which cannot be captured by a finite number of axioms. Nor do I understand how it can comprehend Franzen's proof of ZFC consistency.

>
>
>
> > > You should take a look at Geoffrey Hellman's "Mathematics Without Numbers".
>
> >
>
> > I would be interested what you had to say about this, not some third party.
>
> >
>
>
>
> Well, the basic idea is that mathematical assertions can be re-interpreted in a language with modal operators and second-order quantifiers. The question of how we come to have knowledge of the mathematical assertions is an interesting one, but you could maybe construct some kind of story about it based on our experience with actual finite structures and our intuitions about logical possibilities.
>
>
>
> It's a bit difficult to sum up all in one go. I would suggest that you read the book and then get back to me.
>
>
>
> > > In the truth predicate.
>
> >
>
> > How do you express "Axiom A1 is manifestly true" in ZFC*?
>
> >
>
>
>
> Apply the truth predicate to the Goedel code for axiom A1.

T(A1) is an assertion that A1 is true. The assertion can be either true or false. Franzen says that A1 is manifestly true; A1 cannot be false.

>
>
>
> > > Yes, but in this case the falsity of the hypothesis is not provable in the classical predicate calculus. You need some nonlogical axioms of number theory to get it.
>
> >
>
> > (x)((x#x) -> (x = x+4))
>
> >
>
> > Not sure what you are trying to accomplish but this nitpicking. The point is that classical predicate calculus induces this kind of spurious "truths".
>
> >
>
>
>
> Right, so why is that not manifestly true?

It is counterintuitive. The undergraduate students do not understand it. Only when they are properly schooled in empty sets does it become manifestly true to them. Franzen defines "manifestly true" as utterly compelling, obvious.

>
>
>
> > > Well, surely it is manifestly true on the proposed meaning for -> .
>
> >
>
> > You are confusing things quite badly here. If you define Boolean +, ., ~ the usual way then it is true that
>
> >
>
> > ~(A & ~A) v B (1)
>
> >
>
> > is always one. That does NOT mean that
>
> >
>
> > "If it rains and it does not rain then it is sunny"
>
> >
>
> > is true. You cannot define truth arbitrarily.
>
> >
>
>
>
> Why isn't it true?

Because nothing can be predicated about an impossible situation.

>
>
>
> >
>
> >
>
> > BTW, that (1) is always 1 is true is analytic, and I do not think that is what Franzen means by manifest truth.
>
> >
>
> >
>
> >
>
> > Truth means correspondence with reality. So if there is no king of Switzerland on the list of the wise people then you cannot say that "all the kings of Switzerland have been wise" is true.
>
>
>
> Why not?

I guess we will agree that "the 1905 king of Switzerland was wise" is not true. The same can be said about any sentence "the time t king of Switzerland was wise." So "any king of Switzerland was wise" is not true.

>
>
>
> > The same applies to observations on N. But if you do do such silly things then you *will* have problems with pi_1 sentences.
>
>
>
> You still haven't given me an example of a pi_1 sentence which is true on the standard semantics but not true on your semantics.

I gave you three. Here is one of them again:
(x)(y)~[(x + y < 6) & (y = 8)] (2)
Pick y = 8. Then (x)~[(x + 8 < 6) & (8 = 8)] is not truth-relevant because '(x + 8 < 6)' determines the truth value of 2 regardless of the truth value of '(8 = 8)'.

[fom]

Well, then it is a Newberry truth-value gap semantics.

Have you posted it?

[Rupert]

On Friday, May 3, 2013 5:27:58 AM UTC+2, Newberry wrote:
> On Thursday, May 2, 2013 10:08:56 AM UTC-7, Rupert wrote:
>
> > On Thursday, May 2, 2013 4:01:10 PM UTC+2, Newberry wrote:
>
> >
>
> > > On Wednesday, May 1, 2013 10:17:41 AM UTC-7, Rupert wrote:
>
> >
>
> > >
>
> >
>
> > > > On Tuesday, April 30, 2013 4:00:47 AM UTC+2, Newberry wrote:
>
> >
>
> > >
>
> >
>
> > > >
>
> >
>
> > >
>
> >
>
> > > > > On Monday, April 29, 2013 12:29:33 AM UTC-7, Rupert wrote:
>
> >
>
> > >
>
> >
>
> > > > No.
>
> >
>
> > >
>
> >
>
> > > You lost me.
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > I never said anything which in any way implied that the brain processes can't be captured by a Turing machine. I suggested that maybe they can be captured by a finite number of axioms or axiom schemata. If anything this would help the idea that they could be captured by a Turing machine.
>
> Then I do not know how the brain can fathom the structure of natural numbers N, which cannot be captured by a finite number of axioms. Nor do I understand how it can comprehend Franzen's proof of ZFC consistency.
>

Well,, would you agree that it manages to do it somehow?

I mean, I don't really claim to know whether or not the brain could be simulated by a Turing machine, but I would suggest that our best physics gives us pretty good reason to think that that's the case.

What exactly is the problem about "fathoming" the structure of natural numbers N? Do you mean the problem of how we can succeed in uniquely referring to such a structure? Remember that the structure *is* finitely axiomatizable if you allow second-order languages. However, the set of second-order validities is not recursively enumerable.

And I don't understand what the big deal is supposed to be about understanding Franzen's consistency proof. Seems pretty easy to understand to me.

> >
>
> >
>
> >
>
> > > > You should take a look at Geoffrey Hellman's "Mathematics Without Numbers".
>
> >
>
> > >
>
> >
>
> > > I would be interested what you had to say about this, not some third party.
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > Well, the basic idea is that mathematical assertions can be re-interpreted in a language with modal operators and second-order quantifiers. The question of how we come to have knowledge of the mathematical assertions is an interesting one, but you could maybe construct some kind of story about it based on our experience with actual finite structures and our intuitions about logical possibilities.
>
> >
>
> >
>
> >
>
> > It's a bit difficult to sum up all in one go. I would suggest that you read the book and then get back to me.
>
> >
>
> >
>
> >
>
> > > > In the truth predicate.
>
> >
>
> > >
>
> >
>
> > > How do you express "Axiom A1 is manifestly true" in ZFC*?
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > Apply the truth predicate to the Goedel code for axiom A1.
>
> T(A1) is an assertion that A1 is true. The assertion can be either true or false. Franzen says that A1 is manifestly true; A1 cannot be false.
>

The force of the "manifestly" is just a comment about the strength of the intuition that the axioms are true.

> >
>
> >
>
> >
>
> > > > Yes, but in this case the falsity of the hypothesis is not provable in the classical predicate calculus. You need some nonlogical axioms of number theory to get it.
>
> >
>
> > >
>
> >
>
> > > (x)((x#x) -> (x = x+4))
>
> >
>
> > >
>
> >
>
> > > Not sure what you are trying to accomplish but this nitpicking. The point is that classical predicate calculus induces this kind of spurious "truths".
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > Right, so why is that not manifestly true?
>
> It is counterintuitive. The undergraduate students do not understand it. Only when they are properly schooled in empty sets does it become manifestly true to them. Franzen defines "manifestly true" as utterly compelling, obvious.
>

Well, the connective -> means material implication, certainly you need to be taught what that means before you can judge whether statements involving the connective are "manifestly true" or not.

> >
>
> >
>
> >
>
> > > > Well, surely it is manifestly true on the proposed meaning for -> .
>
> >
>
> > >
>
> >
>
> > > You are confusing things quite badly here. If you define Boolean +, ., ~ the usual way then it is true that
>
> >
>
> > >
>
> >
>
> > > ~(A & ~A) v B (1)
>
> >
>
> > >
>
> >
>
> > > is always one. That does NOT mean that
>
> >
>
> > >
>
> >
>
> > > "If it rains and it does not rain then it is sunny"
>
> >
>
> > >
>
> >
>
> > > is true. You cannot define truth arbitrarily.
>
> >
>
> > >
>
> >
>
> >
>
> >
>
> > Why isn't it true?
>
> Because nothing can be predicated about an impossible situation.
>

Why not?

> >
>
> >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > BTW, that (1) is always 1 is true is analytic, and I do not think that is what Franzen means by manifest truth.
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > >
>
> >
>
> > > Truth means correspondence with reality. So if there is no king of Switzerland on the list of the wise people then you cannot say that "all the kings of Switzerland have been wise" is true.
>
> >
>
> >
>
> >
>
> > Why not?
>
> I guess we will agree that "the 1905 king of Switzerland was wise" is not true. The same can be said about any sentence "the time t king of Switzerland was wise." So "any king of Switzerland was wise" is not true.
>

Sounds fine to me. But we were talking about a sentence using the word "all".

> >
>
> >
>
> >
>
> > > The same applies to observations on N. But if you do do such silly things then you *will* have problems with pi_1 sentences.
>
> >
>
> >
>
> >
>
> > You still haven't given me an example of a pi_1 sentence which is true on the standard semantics but not true on your semantics.
>
>
>
> I gave you three. Here is one of them again:
>
> (x)(y)~[(x + y < 6) & (y = 8)] (2)
>
> Pick y = 8. Then (x)~[(x + 8 < 6) & (8 = 8)] is not truth-relevant because '(x + 8 < 6)' determines the truth value of 2 regardless of the truth value of '(8 = 8)'.

Seems to me you need to give a formal definition of this notion of "determining the truth-value". Do you do that in your paper?

[Nam Nguyen]

Your formula begins with En. How would En express "There's _no_ n ...."?

[Nam Nguyen]

Also, are you sure this is about "counter examples of Goldbach
Conjecture" (your "&p+q=m")?

[Rupert]

Suppoose that n=4. Then the formula says that for all larger m, either m is odd or m is a sum of two primes. This is the same as saying that there are no counter-examples to the Goldbach conjecture.

[Nam Nguyen]

But suppose there are _only_ 10 counter examples (which is still a
possibility), finitely many for sure, how would the formula express
that?