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.

[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?

[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

[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

[scattered]

So it seems that perhaps he didn't make the claim regarding ZFC but only regarding the weaker system of PA; which is hardly surprising since PA is generally considered to be nonproblematic in a way that ZFC is not. Of course strict finitists such as Edward Nelson might be unconvinced, but such skeptics are in a small minority.

[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 10:37 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.
>


--

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

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

[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.

[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?

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[Frederick Williams]

Nam Nguyen wrote:
>
> On 26/04/2013 11:09 AM, Nam Nguyen wrote:

> > On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
> >>
> >> Now PA has been proved consistent in ZF or NBG, but then that
> >> brings the consistency of axioms for set theory.
>
> Exactly right. And exactly my point.
>
> Somewhere, somehow, a circularity or an infinite regression
> of _mathematical knowledge_ will be reached,

How does one reach an infinite regression?

> and at that point
> we still have to confront with the issue of mathematical relativity.

It is not the case that either we go round in a circle or we regress
forever.

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

--

When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[Nam Nguyen]

On 04/05/2013 10:07 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 26/04/2013 11:09 AM, Nam Nguyen wrote:
>
>>> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
>>>>
>>>> Now PA has been proved consistent in ZF or NBG, but then that
>>>> brings the consistency of axioms for set theory.
>>
>> Exactly right. And exactly my point.
>>
>> Somewhere, somehow, a circularity or an infinite regression
>> of _mathematical knowledge_ will be reached,
>
> How does one reach an infinite regression?

By claiming that the state of consistency of PA can be
proved _IN_ a _different formal system_ .

>
>> and at that point
>> we still have to confront with the issue of mathematical relativity.
>
> It is not the case that either we go round in a circle or we regress
> forever.

That's not a refute. Of course.

(It's just an unsubstantiated claim).

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


--

[fom]

On 5/4/2013 11:07 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 26/04/2013 11:09 AM, Nam Nguyen wrote:
>
>>> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
>>>>
>>>> Now PA has been proved consistent in ZF or NBG, but then that
>>>> brings the consistency of axioms for set theory.
>>
>> Exactly right. And exactly my point.
>>
>> Somewhere, somehow, a circularity or an infinite regression
>> of _mathematical knowledge_ will be reached,
>
> How does one reach an infinite regression?
>
>> and at that point
>> we still have to confront with the issue of mathematical relativity.
>
> It is not the case that either we go round in a circle or we regress
> forever.

Out of curiosity, how do you come to that conclusion? I have
come to the exact opposite conclusion. The only sense I can
make of foundations is that it is more like a jigsaw puzzle
that must address circularity and regress directly and with
the objective of making it harmless.

[Nam Nguyen]

I couldn't agree less; and that is exactly what I've proposed
for the past years.

The way out of, or the way to manage and address, the circularity or
and the regression mentioned above is to accept certain non-inference
rules about unknowability, impossibility, as part of the FOL reasoning
edifice.

The entire human mathematical reasoning would then be a balance of
what we can know, through the canonical rules of inference, and what
we can _not_ know, through the newly accepted _non-inference rules_ .

Exactly what these rules of non-inference are we can sort it out.
But there's a consequence we have to accept as well: mathematics
in general would be relativistic, with certain mathematical truth
values can be chosen _by choice_ (at will).

[fom]

On 5/5/2013 12:52 AM, Nam Nguyen wrote:
> On 04/05/2013 6:04 PM, fom wrote:
>> On 5/4/2013 11:07 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>>
>>>> On 26/04/2013 11:09 AM, Nam Nguyen wrote:
>>>
>>>>> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
>>>>>>
>>>>>> Now PA has been proved consistent in ZF or NBG, but then that
>>>>>> brings the consistency of axioms for set theory.
>>>>
>>>> Exactly right. And exactly my point.
>>>>
>>>> Somewhere, somehow, a circularity or an infinite regression
>>>> of _mathematical knowledge_ will be reached,
>>>
>>> How does one reach an infinite regression?
>>>
>>>> and at that point
>>>> we still have to confront with the issue of mathematical relativity.
>>>
>>> It is not the case that either we go round in a circle or we regress
>>> forever.
>>
>> Out of curiosity, how do you come to that conclusion? I have
>> come to the exact opposite conclusion.
>
>
>> The only sense I can
>> make of foundations is that it is more like a jigsaw puzzle
>> that must address circularity and regress directly and with
>> the objective of making it harmless.
>
> I couldn't agree less; and that is exactly what I've proposed
> for the past years.
>

Well, I will not dispute your opinions.

I simply know what I have done and how it fits
in with established literature. I also know how
it is non-standard and, thereby, easily subject to
criticism.

[Nam Nguyen]

Yes. Non-standard and non-canonical is a lightning rod
to criticism. But more often than not non-standard
has its past and criticism would be gone.

[Frederick Williams]

Nam Nguyen wrote:
>
> On 04/05/2013 10:07 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> On 26/04/2013 11:09 AM, Nam Nguyen wrote:
> >
> >>> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
> >>>>
> >>>> Now PA has been proved consistent in ZF or NBG, but then that
> >>>> brings the consistency of axioms for set theory.
> >>
> >> Exactly right. And exactly my point.
> >>
> >> Somewhere, somehow, a circularity or an infinite regression
> >> of _mathematical knowledge_ will be reached,
> >
> > How does one reach an infinite regression?
>
> By claiming that the state of consistency of PA can be
> proved _IN_ a _different formal system_ .

Your notion of infinite is very modest if does not go beyond two.

> >
> >> and at that point
> >> we still have to confront with the issue of mathematical relativity.
> >
> > It is not the case that either we go round in a circle or we regress
> > forever.
>
> That's not a refute. Of course.
>
> (It's just an unsubstantiated claim).

And yet an obviously true one. Suppose the question of the consistency
of PA is raised, a party to the discussion may say 'I accept that PA is
consistent and I feel no need to prove it.' No circle, no regression.

[Frederick Williams]

fom wrote:
>
> On 5/4/2013 11:07 AM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >> On 26/04/2013 11:09 AM, Nam Nguyen wrote:
> >
> >>> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
> >>>>
> >>>> Now PA has been proved consistent in ZF or NBG, but then that
> >>>> brings the consistency of axioms for set theory.
> >>
> >> Exactly right. And exactly my point.
> >>
> >> Somewhere, somehow, a circularity or an infinite regression
> >> of _mathematical knowledge_ will be reached,
> >
> > How does one reach an infinite regression?
> >
> >> and at that point
> >> we still have to confront with the issue of mathematical relativity.
> >
> > It is not the case that either we go round in a circle or we regress
> > forever.
>
> Out of curiosity, how do you come to that conclusion?

I was thinking of the question of PA's consistency. If someone just
accepts it, then he neither goes in a circle nor does he regress
forever.

> I have
> come to the exact opposite conclusion. The only sense I can
> make of foundations is that it is more like a jigsaw puzzle
> that must address circularity and regress directly and with
> the objective of making it harmless.

[fom]

On 5/5/2013 9:47 AM, Frederick Williams wrote:
> fom wrote:
>>
>> On 5/4/2013 11:07 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>>
>>>> On 26/04/2013 11:09 AM, Nam Nguyen wrote:
>>>
>>>>> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
>>>>>>
>>>>>> Now PA has been proved consistent in ZF or NBG, but then that
>>>>>> brings the consistency of axioms for set theory.
>>>>
>>>> Exactly right. And exactly my point.
>>>>
>>>> Somewhere, somehow, a circularity or an infinite regression
>>>> of _mathematical knowledge_ will be reached,
>>>
>>> How does one reach an infinite regression?
>>>
>>>> and at that point
>>>> we still have to confront with the issue of mathematical relativity.
>>>
>>> It is not the case that either we go round in a circle or we regress
>>> forever.
>>
>> Out of curiosity, how do you come to that conclusion?
>
> I was thinking of the question of PA's consistency. If someone just
> accepts it, then he neither goes in a circle nor does he regress
> forever.
>

That is probably "standard mathematics". In a thread discussing
belief and proof I pointed out that there can be second-order
consequences that cannot be proven in a sound deductive system.
That may be an analog to the notion of true justified belief since
it is related to consequence but cannot be adequately proven.

I suppose I can place your statement into that kind of context
unproblematically.

What I would like to think is that there is an implicit circularity
if one takes belief out of the picture.

Thanks.

[Nam Nguyen]

On 05/05/2013 8:45 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>> On 04/05/2013 10:07 AM, Frederick Williams wrote:
>>> Nam Nguyen wrote:
>>>>
>>>> On 26/04/2013 11:09 AM, Nam Nguyen wrote:
>>>
>>>>> On 2013-04-25, FredJeffries <fredje...@gmail.com> wrote:
>>>>>>
>>>>>> Now PA has been proved consistent in ZF or NBG, but then that
>>>>>> brings the consistency of axioms for set theory.
>>>>
>>>> Exactly right. And exactly my point.
>>>>
>>>> Somewhere, somehow, a circularity or an infinite regression
>>>> of _mathematical knowledge_ will be reached,
>>>
>>> How does one reach an infinite regression?
>>
>> By claiming that the state of consistency of PA can be
>> proved _IN_ a _different formal system_ .
>
> Your notion of infinite is very modest if does not go beyond two.

That does _not_ mean there be only two, actually.

>
>>>
>>>> and at that point
>>>> we still have to confront with the issue of mathematical relativity.
>>>
>>> It is not the case that either we go round in a circle or we regress
>>> forever.
>>
>> That's not a refute. Of course.
>>
>> (It's just an unsubstantiated claim).
>
> And yet an obviously true one. Suppose the question of the consistency
> of PA is raised, a party to the discussion may say 'I accept that PA is
> consistent and I feel no need to prove it.' No circle, no regression.

The circularity rests with the argument on the _actual and objective_
state of consistency of PA, _not_ on the _wishful and subjective_
"acceptance" of anything.

[Frederick Williams]

Mathematicians (like the rest of humanity) are forever accepting
things. It is no big deal.

[Nam Nguyen]

Verification, proving, is a big deal.

[Nam Nguyen]

For example, would you _accept_ the consistency of PA + ~cGC
("It is no big deal" you said)?

[Frederick Williams]

Not everyone shares your obsessions.

The consistency of PA may be an objective fact (or fiction), but proving
is a human activity.

>

[Frederick Williams]

Nam Nguyen wrote:

>
> The circularity rests with the argument on the _actual and objective_
> state of consistency of PA, _not_ on the _wishful and subjective_
> "acceptance" of anything.

He who argues probably accepts his own argument, and he argues probably
because he wishes others to accept it.

--
I think I am an Elephant,
Behind another Elephant
Behind /another/ Elephant who isn't really there....
A.A. Milne, Now we are six

[Nam Nguyen]

Well then prove to the fora that cGC is true in the naturals, or that
~cGC is.

Isn't it true that Torkel Franzen once alluded to a similar notion that
one could "prove" the consistency of PA simply by observing that PA
wouldn't prove the false statement '0=1'?

But where would the false statement here: cGC, or ~cGC?