Re: The Invalidity of Godel's Incompleteness Work.

[Peter Percival]

Marshall wrote:
> On Wednesday, November 20, 2013 8:56:48 AM UTC-8, Peter Percival wrote:
>>
>> Madam Life's a piece in bloom,
>> Death goes dogging everywhere:
>> She's the tenant of the room,
>> He's the ruffian on the stair.
>
> What is that, may I ask? It's delightful. Morbid yet upbeat.
> I love the how dactyl "ruffian" is thrown in on the last line
> to disturb the regularity of all the trochees.
>
>
> Marshall

It is from a poem by William Ernest Henley. Henley had a leg amputated
and is the person referred to by Alan Smail. I first came across /The
ruffian on the stair/ as the title of a play by Joe Orton.

--
Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.

[Nam Nguyen]

On 21/11/2013 4:09 AM, Rupert wrote:
> On Thursday, November 21, 2013 8:23:05 AM UTC+1, Nam Nguyen wrote:
>> On 21/11/2013 12:12 AM, Rupert wrote:
>>> On Thursday, November 21, 2013 7:22:08 AM UTC+1, Nam Nguyen wrote:
>>>> On 20/11/2013 6:22 AM, Rupert wrote:
>>>>> No. You have not demonstrated this.
>>
>>>> I did prove the invalidity .
>>
>>> You haven't done any such thing.
>>
>> I did _repeatedly_ .
>>
>
> You believe that you did. But, in fact, the argument you offered is worthless.

This is the 2nd times you, as a self-proclaimed informed mathematician,
_initiated the acidity_ (your "worthless") into a foundation (technical)
argument.

You're of the freaking pathetic Inquisition mentality, who isn't arguing
in good faith.

>>>> I did show "(1) and (2)" is an invalid
>>>> hypothesis mainly because the consistency of a theory T can't be
>>>> proven, hence it's invalid to hypothesize (1) and (2) wouldn't lead
>>>> to a contradiction.
>>
>>> When you make the assertion "the consistency of a theory T can't be proven", do you understand that you need to specify what methods of proof are allowed?
>>
>> I did: methods of proof (in meta level) would involve permissible
>> definitions of consistency (amongst other things).
>>
>
> You are making a claim that you specified what methods of proof are allowed. However, you didn't do that just now, in the sentence immediately above. You made a vague statement about what the methods of proof would involve, and then you said "amongst other things", thereby indicating that there are other things that the methods of proof would involve which you haven't told us about. So you haven't specified what methods of proof are allowed in the sentence immediately above. You seem to want to claim that you have done this elsewhere. But I have no memory of your having done so. I could go back and re-read all your posts to see if I can find the place where you specified what methods of proof are allowed, if I felt so inclined. I have a feeling that this wouldn't get me very far.

I did explain that more than one time before: the permissible
definitions and reasoning methods within FOL(=); the list is finite
and we don't have to list them all here in this argument: you don't
seem to even understand what the permissible definitions for consistency
are, let alone me wasting time listing other _relevant_ ones!
>
>>>> To counter my argument, you claimed Q is consistent as a counter
>>>> example, but have failed to prove your claim using _valid definition_
>>>> _of consistency_ .
>>
>>> That's false. Shoenfield's book has the proof that Q is consistent, and there's nothing wrong with the definition of consistency that he uses.
>>
>> But didn't you say PRA was used there?
>
> The argument can be formalised as a formal proof in PRA.

But is PRA consistent? And does the definition of consistency
of a general T say you could formalize the proof of Q's consistency
in _another axiom-set_ (PRA) ?

Why do you keep ignoring the below fact:

>> _The definition of consistency or inconsistency does NOT provide a_
>> _secondary back up system for proving consistency or inconsistency_ .

?

>
>> If so that's not conforming to
>> the definition of consistency applied to Q, since PRA is a different
>> axiom-set (different theory) than Q.
>>
>
> Well, I don't know why you would think that. Why is it not conforming to the definition of consistency?

This is why:

>> _The definition of consistency or inconsistency does NOT provide a_
>> _secondary back up system for proving consistency or inconsistency_ .

Shoenfield uses the same definition as you.

I already explained to you he was _glossing_ , writing for educational
purposes. His book isn't suitable to examine deeper issues at
foundation.

>
>> _I already did address this point_ .
>>
>
> You've never given any real reason for thinking that Shoenfield's proof doesn't conform to the definition of consistency.

I already did: you just pathetically ignored my reasons.

>>
>>> Also, even if it were so, it wouldn't follow that G�del's incompleteness theorem is invalid.
>>
>> I did, via my definition of meta level invalid inference.
>>
>> Do you understand this definition (and the lottery analogy)?
>>
>
> I can't remember you giving this definition, and I don't see the point of the lottery analogy at all.
>
That's why your arguments were/are pathetic: full of forgetfulness,
ignoring, and yet full of ad hominem attacks.

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

NYOGEN SENZAKI

[fom]

On 11/21/2013 5:18 AM, Alan Smaill wrote:
> fom <fom...@nyms.net> writes:
>
>> Nam appears to be "ineducable" on these matters. So, if one
>> wishes to extend a certain graciousness toward his statements,
>> then one needs to interpret his statements at face value under
>> the presumption that they reflect his beliefs.
>
> Fine.
> But I am immediately reminded of the occasional attempts to
> support WM's views, which WM either totally ignores,
> or rejects outright.
>
>> My remarks arise from the fact that I have thought about
>> his presentations without giving credence to the standard
>> interpretations for his words. At their core are constructive
>> notions of feasibility and verification.
>
> This is not my impression;
> he *might* be equating *knowability* with truth plus some other
> criterion not yet clarified; AIUI he's not advocating a different
> notion of truth itself as intuitionists do.
>
> But Nam could tell us himself, I presume.
>
>

As you said the other day: I wish.

[Nam Nguyen]

On 21/11/2013 4:35 AM, Alan Smaill wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> On 20/11/2013 3:43 AM, Alan Smaill wrote:
>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>
>>>> The other half is demonstrating why Godel Completeness Theorem is
>>>> invalid, which now we'd do.
>>>>
>>>> In a modern form, the Completeness Theorem asserts the following
>>>> equivalence:
>>>>
>>>> (T has a model) <=> (T is consistent).
>>>>
>>>> But by definition, a model of T is a language structure M where all
>>>> the statement-theorems of T are true, meaning that _this already_
>>>> _presupposes T is consistent_ .
>>>
>>> What are you trying to say?
>>
>> Here's the definition of T having M as its model:
>>
>> (M is a model of T) df= (M is a structure) and
>> (T is consistent) and
>> ((S is a statement-theorem in T) => (M |= S)).
>
> That's *not* what everyone else uses as definition of model.

You don't have a proof for that. The only thing it matters
is whether or not it's a technically correct definition,
and it is.

> Look at Shoenfield, for once.

_You_ should look it at least more twice: his definition
contains a circularity. And in any rate from there you can
conclude my there's nothing technically wrong with my definition above.

>
> In your terms, here's something nore like the normal definition:
>
> (M is a model of T) df= (M is a structure) and
> ((S is a statement-theorem in T) => (M |= S)).
>
>
>> This definition presupposes T is consistent, because otherwise ~Ax[x=x]
>> would be true in M, a violation of language structure definition.
>
> If you are using the definition above, it is *you* that introduced
> the circularity.
>
>>> The equivalence claim applies even if T is inconsistent.
>>
>> Let's find a closure on my "this already presupposes T is consistent",
>> as shown above, first.
>
> My experience tells me that "closure then will address" is
> equivalent to "closure then will ignore" in your hands.

My experience tells me that you and your side has a long history
of resisting facts, valid definition, valid arguments, in this
kind of debates.

>
> But please do address the issue.

I've addressed the issue with Shoenfield right above. Why don't
you just review what I've just said and let me know specifically
what's wrong with my definition of model of a T.

And I mean "specifically", not innuendo, or "blah blah ..." kind
of arguments.

[Nam Nguyen]

On 21/11/2013 4:28 AM, Alan Smaill wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>

>> On 20/11/2013 3:38 AM, Alan Smaill wrote:
>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>

>>>> Do you _understand_ my definition of "invalid" meta inference
>>>> here? (Note: I'm _not_ asking if you'd agree with it).
>>>
>>> No, I find it confusing.
>>
>> My definition is only a mere few sentence long. Which part
>> of it did you find confusing? Or do you even remember the
>> definition?
>>
>> Just say "I find it confusing" doesn't mean anything in a
>> debate.
>
> I'm just answering your question.

So, Alan couldn't answer a straight Yes, or No?
>
> Now you know the answer,
>
In any rate, _specifically why_ did you find my definition
confusing?

[Nam Nguyen]

I already did define what it means by "It's impossible to know"
in this context _multiple times_ .

That you, fom, or anyone else _still don't understand_ isn't my problem.

[Nam Nguyen]

On 21/11/2013 4:11 AM, Rupert wrote:
> On Thursday, November 21, 2013 8:23:28 AM UTC+1, Marshall wrote:
>>
>> So Rupert. Now it is possible to see clearly the sort of person you're arguing
>> with: someone who dismisses your disproof of his claim as being merely
>> "some sort of counterexample" and follows it up by exhorting you to
>> "please remember things straight."
>>
>> Consider how much progress you're going to make discussing things
>> with someone who sees counterexamples to universal statements
>> as irrelevant. Now think about how many years you've been arguing
>> with Nam; how many hours spent at the keyboard.
>>
>> Now ask yourself why.
>>
>
> Because I'm feeling the need to take a break from reading about semisimple algebraic groups, and there's nothing else to do, and it's mildly diverting.

Freaking pathetic! Liar!

[Rupert]

On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
> On 21/11/2013 4:09 AM, Rupert wrote:
> > On Thursday, November 21, 2013 8:23:05 AM UTC+1, Nam Nguyen wrote:
> >> On 21/11/2013 12:12 AM, Rupert wrote:
> >>> On Thursday, November 21, 2013 7:22:08 AM UTC+1, Nam Nguyen wrote:
> >>>> On 20/11/2013 6:22 AM, Rupert wrote:
> >>>>> No. You have not demonstrated this.
>
> >>>> I did prove the invalidity .
>
> >>> You haven't done any such thing.
>
> >> I did _repeatedly_ .
>
> > You believe that you did. But, in fact, the argument you offered is worthless.
>
> This is the 2nd times you, as a self-proclaimed informed mathematician,
> _initiated the acidity_ (your "worthless") into a foundation (technical)
> argument.
>

I'm not really completely clear what this sentence means.

> You're of the freaking pathetic Inquisition mentality, who isn't arguing
> in good faith.
>

This accusation lacks a rational foundation.

> >>>> I did show "(1) and (2)" is an invalid
> >>>> hypothesis mainly because the consistency of a theory T can't be
> >>>> proven, hence it's invalid to hypothesize (1) and (2) wouldn't lead
> >>>> to a contradiction.
>
> >>> When you make the assertion "the consistency of a theory T can't be proven", do you understand that you need to specify what methods of proof are allowed?
>
> >> I did: methods of proof (in meta level) would involve permissible
> >> definitions of consistency (amongst other things).
>
> > You are making a claim that you specified what methods of proof are allowed. However, you didn't do that just now, in the sentence immediately above. You made a vague statement about what the methods of proof would involve, and then you said "amongst other things", thereby indicating that there are other things that the methods of proof would involve which you haven't told us about. So you haven't specified what methods of proof are allowed in the sentence immediately above. You seem to want to claim that you have done this elsewhere. But I have no memory of your having done so. I could go back and re-read all your posts to see if I can find the place where you specified what methods of proof are allowed, if I felt so inclined. I have a feeling that this wouldn't get me very far.
>
> I did explain that more than one time before: the permissible
> definitions and reasoning methods within FOL(=); the list is finite
> and we don't have to list them all here in this argument: you don't
> seem to even understand what the permissible definitions for consistency
> are, let alone me wasting time listing other _relevant_ ones!
>

Well, I certainly don't know what you mean by "the permissible definitions within FOL(=)". If the allowable methods of proof are the reasoning methods within FOL(=), then what nonlogical axioms do you allow?

> >>>> To counter my argument, you claimed Q is consistent as a counter
> >>>> example, but have failed to prove your claim using _valid definition_
> >>>> _of consistency_ .
>
> >>> That's false. Shoenfield's book has the proof that Q is consistent, and there's nothing wrong with the definition of consistency that he uses.
>
> >> But didn't you say PRA was used there?
>
> > The argument can be formalised as a formal proof in PRA.
>
> But is PRA consistent?

Yes.

> And does the definition of consistency
> of a general T say you could formalize the proof of Q's consistency
> in _another axiom-set_ (PRA) ?
>

Why would there be a problem with doing that?

> Why do you keep ignoring the below fact:
>
> >> _The definition of consistency or inconsistency does NOT provide a_
> >> _secondary back up system for proving consistency or inconsistency_ .

Because I have no idea what it's supposed to mean.

> ?
>
> >> If so that's not conforming to
> >> the definition of consistency applied to Q, since PRA is a different
> >> axiom-set (different theory) than Q.
>
> > Well, I don't know why you would think that. Why is it not conforming to the definition of consistency?
>
> This is why:
>
> >> _The definition of consistency or inconsistency does NOT provide a_
> >> _secondary back up system for proving consistency or inconsistency_ .
>

I am sorry, I cannot follow your reasoning.

> Shoenfield uses the same definition as you.
>
> I already explained to you he was _glossing_ , writing for educational
> purposes. His book isn't suitable to examine deeper issues at
> foundation.
>

The proof he gives is fine.

> >> _I already did address this point_ .
>
> > You've never given any real reason for thinking that Shoenfield's proof doesn't conform to the definition of consistency.
>
> I already did: you just pathetically ignored my reasons.
>

No, actually, my remark was correct.

> >>> Also, even if it were so, it wouldn't follow that G�del's incompleteness theorem is invalid.

>
> >> I did, via my definition of meta level invalid inference.
>
> >> Do you understand this definition (and the lottery analogy)?
>
> > I can't remember you giving this definition, and I don't see the point of the lottery analogy at all.
>
> That's why your arguments were/are pathetic: full of forgetfulness,
> ignoring, and yet full of ad hominem attacks.
>

Right. So what do you suppose is the best way forward? Will you keep going in the hope that I will improve?

[Nam Nguyen]

On 21/11/2013 10:41 AM, Rupert wrote:
> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:

>> This is why:
>>
>> >> _The definition of consistency or inconsistency does NOT provide a_
>> >> _secondary back up system for proving consistency or inconsistency_ .
>>
>
> I am sorry, I cannot follow your reasoning.

Which part of my statement above isn't clear to you.

Can you cite standard textbook FOL(=) definitions
of inconsistency and consistency? Then I'll explain
my statement more to you.

>>
>
> Right. So what do you suppose is the best way forward? Will you keep going in the hope that I will improve?
>

Can you cite standard textbook FOL(=) definitions
of inconsistency and consistency?

Then I'll explain my statement more to you, to move forward
the argument.

[Nam Nguyen]

On 21/11/2013 10:48 AM, Nam Nguyen wrote:
> On 21/11/2013 10:41 AM, Rupert wrote:
>> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>
>>> This is why:
>>>
>>> >> _The definition of consistency or inconsistency does NOT provide a_
>>> >> _secondary back up system for proving consistency or
>>> inconsistency_ .
>>>
>>
>> I am sorry, I cannot follow your reasoning.
>

> Which part of my statement above isn't clear to you?

>
> Can you cite standard textbook FOL(=) definitions
> of inconsistency and consistency? Then I'll explain
> my statement more to you.
>
>>>
>>
>> Right. So what do you suppose is the best way forward? Will you keep
>> going in the hope that I will improve?
>>
>
> Can you cite standard textbook FOL(=) definitions
> of inconsistency and consistency?
>
> Then I'll explain my statement more to you, to move forward
> the argument.

Cite the requested definitions, if you genuinely desire to technically
refute my arguments about the invalidity of Godel's Incompleteness work.

Everything hings on these 2 simple and standard definitions!

[Nam Nguyen]

On 21/11/2013 11:21 AM, Nam Nguyen wrote:
> On 21/11/2013 10:48 AM, Nam Nguyen wrote:
>> On 21/11/2013 10:41 AM, Rupert wrote:
>>> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>>
>>>> This is why:
>>>>
>>>> >> _The definition of consistency or inconsistency does NOT
>>>> provide a_
>>>> >> _secondary back up system for proving consistency or
>>>> inconsistency_ .
>>>>
>>>
>>> I am sorry, I cannot follow your reasoning.
>>
>> Which part of my statement above isn't clear to you?
>>
>> Can you cite standard textbook FOL(=) definitions
>> of inconsistency and consistency? Then I'll explain
>> my statement more to you.
>>
>>>>
>>>
>>> Right. So what do you suppose is the best way forward? Will you keep
>>> going in the hope that I will improve?
>>>
>>
>> Can you cite standard textbook FOL(=) definitions
>> of inconsistency and consistency?
>>
>> Then I'll explain my statement more to you, to move forward
>> the argument.
>
> Cite the requested definitions, if you genuinely desire to technically
> refute my arguments about the invalidity of Godel's Incompleteness work.
>
> Everything hings on these 2 simple and standard definitions!

I meant "Everything hinges on these 2 simple and standard definitions!"

[Nam Nguyen]

On 21/11/2013 10:09 AM, Nam Nguyen wrote:
> On 21/11/2013 4:28 AM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 20/11/2013 3:38 AM, Alan Smaill wrote:
>>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>>
>>>>> Do you _understand_ my definition of "invalid" meta inference
>>>>> here? (Note: I'm _not_ asking if you'd agree with it).
>>>>
>>>> No, I find it confusing.
>>>
>>> My definition is only a mere few sentence long. Which part
>>> of it did you find confusing? Or do you even remember the
>>> definition?
>>>
>>> Just say "I find it confusing" doesn't mean anything in a
>>> debate.
>>
>> I'm just answering your question.
>
> So, Alan couldn't answer a straight Yes, or No?

My apology: You did answer "No".

>>
>> Now you know the answer,
>>
> In any rate, _specifically why_ did you find my definition
> confusing?

Well? _Specifically why_ did you find my definition confusing?

[Julio Di Egidio]

"fom" <fom...@nyms.net> wrote in message
news:RL2dnVTpyoizPBDP...@giganews.com...
> On 11/20/2013 11:33 PM, Marshall wrote:
>>
>> Interesting. I've always felt (somewhat against the current,
>> it seems) that semantics is prior to syntax.
>
> I am of the same feeling.
>
> But, it is in the nature of logical investigation
> that there is a reduction to syntax.
<snip>

> In the case of reductions to syntax, that is where
> we must begin the synthesis for a foundational
> answer.

You could maybe ascribe that to the dumbest logicism, but to claim that that
"is in the nature of logical investigation" is just outright nonsense.

> But, the synthesis is the outcome of analysis.
> If you ignore the circularity of the method, then
> you lose sight of of the problem which is being
> addressed.

Not even wrong.

Julio

[Marshall]

On Thursday, November 21, 2013 9:41:19 AM UTC-8, Rupert wrote:
> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>
> > > You believe that you did. But, in fact, the argument you offered is worthless.
> >
> > This is the 2nd times you, as a self-proclaimed informed mathematician,
> > _initiated the acidity_ (your "worthless") into a foundation (technical)
> > argument.
>
> I'm not really completely clear what this sentence means.

I believe what he means is that your use of the word "worthless"
was bad and you should feel bad.


Marshall

[Nam Nguyen]

On 21/11/2013 10:41 AM, Rupert wrote:
> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>> On 21/11/2013 4:09 AM, Rupert wrote:
>>> On Thursday, November 21, 2013 8:23:05 AM UTC+1, Nam Nguyen wrote:
>>
>>>> But didn't you say PRA was used there?
>>
>>> The argument can be formalised as a formal proof in PRA.
>>
>> But is PRA consistent?
>
> Yes.

But how would you prove it, using FOL(=) definition of consistency?

Or are you just assuming PRA is consistent?

[George Greene]

> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
> > Why do you keep ignoring the below fact:
>
> >
>
> > >> _The definition of consistency or inconsistency does NOT provide a_
>
> > >> _secondary back up system for proving consistency or inconsistency_ .
>
>

On Thursday, November 21, 2013 12:41:19 PM UTC-5, Rupert wrote:
> Because I have no idea what it's supposed to mean.

Don't panic.
Nam has no idea what it means either.

[Nam Nguyen]

Of course I do. Care for citing the standard (syntactical) definition
of consistency here?

[Nam Nguyen]

On 21/11/2013 8:11 PM, Nam Nguyen wrote:
> On 21/11/2013 8:04 PM, George Greene wrote:
>>
>>> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>>>> Why do you keep ignoring the below fact:
>>>
>>>>
>>>
>>>> >> _The definition of consistency or inconsistency does NOT
>>>> provide a_
>>>
>>>> >> _secondary back up system for proving consistency or
>>>> inconsistency_ .
>>>
>>>
>>
>> On Thursday, November 21, 2013 12:41:19 PM UTC-5, Rupert wrote:
>>> Because I have no idea what it's supposed to mean.
>>
>>
>> Don't panic.
>> Nam has no idea what it means either.
>
> Of course I do. Care for citing the standard (syntactical) definition
> of consistency here?

Or inconsistency for that matter.

[Nam Nguyen]

On 21/11/2013 8:13 PM, Nam Nguyen wrote:
> On 21/11/2013 8:11 PM, Nam Nguyen wrote:
>> On 21/11/2013 8:04 PM, George Greene wrote:
>>>
>>>> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>>>>> Why do you keep ignoring the below fact:
>>>>
>>>>
>>>>> >> _The definition of consistency or inconsistency does NOT
>>>>> provide a_
>>>>
>>>>> >> _secondary back up system for proving consistency or
>>>>> inconsistency_ .
>>>>
>>>>
>>>
>>> On Thursday, November 21, 2013 12:41:19 PM UTC-5, Rupert wrote:
>>>> Because I have no idea what it's supposed to mean.
>>>
>>>
>>> Don't panic.
>>> Nam has no idea what it means either.
>>
>> Of course I do. Care for citing the standard (syntactical) definition
>> of consistency here?
>
> Or inconsistency for that matter.

So, everyone is eager to attack either the messenger or his message,
and yet when it comes to the most important (albeit simple) definition
no one cares to say it?

[Marshall]

On Thursday, November 21, 2013 8:41:45 PM UTC-8, Nam Nguyen wrote:
>
> So, everyone is eager to attack either the messenger or his message,
> and yet when it comes to the most important (albeit simple) definition
> no one cares to say it?

One thing that people who aren't acting in good faith do
is get hugely pedantic about trivialities.

Everyone here knows what consistency is. Challenging the
world to come up with a definition for it is arguing over
trivialities instead of arguing about substance. You've
done this enough times that, I expect, no one wants
to take the bait. (I'll know you've missed the point of
my post entirely if you reply with, "Oh, so do you speak
for everyone, Marshall?")

It must be obvious to you that few of the people replying
to you take you at all seriously. (I actually can't think of
any.) If you're happy with how the conversations have been
going, then keep doing what you've been doing, since it's
working for you. However if you'd like to be taken differently
than you have been in the past, (for example, if you'd like to be
taken seriously) then you will necessarily have to change
your behavior.


Marshall

[Marshall]

On Thursday, November 21, 2013 3:11:32 AM UTC-8, Rupert wrote:
> On Thursday, November 21, 2013 8:23:28 AM UTC+1, Marshall wrote:
> >
> > So Rupert. Now it is possible to see clearly the sort of person you're arguing
> > with: someone who dismisses your disproof of his claim as being merely
> > "some sort of counterexample" and follows it up by exhorting you to
> > "please remember things straight."
> >
> > Consider how much progress you're going to make discussing things
> > with someone who sees counterexamples to universal statements
> > as irrelevant. Now think about how many years you've been arguing
> > with Nam; how many hours spent at the keyboard.
> >
> > Now ask yourself why.
> >
>
> Because I'm feeling the need to take a break from reading about
> semisimple algebraic groups, and there's nothing else to do,
> and it's mildly diverting.

Those things are all true (except maybe for the "nothing else to do" one.)
But that doesn't necessarily mean they are the true reason.


Marshall

[Nam Nguyen]

On 11/21/2013 10:20 PM, Marshall wrote:
> On Thursday, November 21, 2013 8:41:45 PM UTC-8, Nam Nguyen wrote:
>>
>> So, everyone is eager to attack either the messenger or his message,
>> and yet when it comes to the most important (albeit simple) definition
>> no one cares to say it?
>
> One thing that people who aren't acting in good faith do
> is get hugely pedantic about trivialities.

Really? I had said a trivial thing:

>>_The definition of consistency or inconsistency does NOT provide a_
>>_secondary back up system for proving consistency or inconsistency_ .

Was Rupert able to understand such trivial information? I guess _not_ ,

since he said:

On Thursday, November 21, 2013 12:41:19 PM UTC-5, Rupert wrote:
> Because I have no idea what it's supposed to mean.

That's why I asked him to cite the definition of consistency
or inconsistency.

>
> Everyone here knows what consistency is.

All right then, everyone here knows I'm right and Rupert is
wrong about my meta theorem that consistency can't be proved in meta
level _USING THE DEFINITION OF CONSISTENCY THAT EVERYONE KNOWS_ !

Good grief!

--

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

NYOGEN SENZAKI

----------------------------------------------------

[Marshall]

On Thursday, November 21, 2013 10:40:01 PM UTC-8, Nam Nguyen wrote:
> On 11/21/2013 10:20 PM, Marshall wrote:
> > On Thursday, November 21, 2013 8:41:45 PM UTC-8, Nam Nguyen wrote:
>
> >> So, everyone is eager to attack either the messenger or his message,
> >> and yet when it comes to the most important (albeit simple) definition
> >> no one cares to say it?
> >
> > One thing that people who aren't acting in good faith do
> > is get hugely pedantic about trivialities.
>
> Really? I had said a trivial thing:
>
> >>_The definition of consistency or inconsistency does NOT provide a_
> >>_secondary back up system for proving consistency or inconsistency_ .
>
> Was Rupert able to understand such trivial information? I guess _not_ ,
> since he said:
>
> On Thursday, November 21, 2013 12:41:19 PM UTC-5, Rupert wrote:
> > Because I have no idea what it's supposed to mean.
>
> That's why I asked him to cite the definition of consistency
> or inconsistency.
>
> >
> > Everyone here knows what consistency is.
>
> All right then, everyone here knows I'm right and Rupert is
> wrong about my meta theorem that consistency can't be proved in meta
> level _USING THE DEFINITION OF CONSISTENCY THAT EVERYONE KNOWS_ !
>
> Good grief!

whoosh

[Rupert]

On Thursday, November 21, 2013 6:48:58 PM UTC+1, Nam Nguyen wrote:
> On 21/11/2013 10:41 AM, Rupert wrote:
> > On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>
> >> This is why:
>
> >> >> _The definition of consistency or inconsistency does NOT provide a_
> >> >> _secondary back up system for proving consistency or inconsistency_ .
>
> > I am sorry, I cannot follow your reasoning.
>
> Which part of my statement above isn't clear to you.
>

Your goal is to show that if I try to prove the consistency of Q from the axioms of PRA, then that somehow involves failing to conform to the definition of consistency, right?

So how have you shown that? Like for example with the phrase "a secondary back up system", what exactly does that mean? I mean, suppose that I tried to prove the consistency of Q from the axioms of Q. Would that mean I was not using "a secondary back up system"? Why would that be, what's the difference?

> Can you cite standard textbook FOL(=) definitions
> of inconsistency and consistency? Then I'll explain
> my statement more to you.
>

Well, I can certainly give you a definition, but if you want me to give you a citation to Shoenfield's definition I'll have to wait until I get home, I'm in the office at the moment.

> > Right. So what do you suppose is the best way forward? Will you keep going in the hope that I will improve?
>
> Can you cite standard textbook FOL(=) definitions
> of inconsistency and consistency?
>

Yes, but if you need an actual page reference then I can't give it to you immediately because I haven't got a mathematical logic textbook here in the office. I guess I could see if they've got Shoenfield in the library. I'll give you a page reference to Shoenfield later on when I get home, if you like.

If you just want me to give you a standard definition based on memory, I can certainly do that easily enough, but an actual citation with a page reference I cannot give you right now, you'll have to wait a little bit.

> Then I'll explain my statement more to you, to move forward
> the argument.
>

Can't wait.

[Rupert]

It may well be that he believes that. I am unclear about what the foundation would be for such a moral view.

[Rupert]

On Friday, November 22, 2013 3:11:06 AM UTC+1, Nam Nguyen wrote:
> On 21/11/2013 10:41 AM, Rupert wrote:
> > On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
> >> On 21/11/2013 4:09 AM, Rupert wrote:
> >>> On Thursday, November 21, 2013 8:23:05 AM UTC+1, Nam Nguyen wrote:
>
> >>>> But didn't you say PRA was used there?
>
> >>> The argument can be formalised as a formal proof in PRA.
>
> >> But is PRA consistent?
>
> > Yes.
>
> But how would you prove it, using FOL(=) definition of consistency?
>
> Or are you just assuming PRA is consistent?
>

I believe that it's consistent. If you're asking "Can you prove it", then that would obviously depend on which axioms you're going to allow me to use. But that's neither here nor there, really, because if you've got genuine doubts about the consistency of PRA then any proof that I can give you is not going to be epistemologically convincing for you.

Do you, in fact, have genuine doubts about the consistency of PRA? Do you regard it as an open question?

I mean, I suppose I can say that in some sense it's an open question whether I'm currently using a computer to communicate with someone on the internet. But I don't really feel any need to entertain serious doubts about the matter. If you do have serious doubts about the consistency of PRA or even the consistency of Q, then that's fine. However, it remains an objective fact that a proof of the consistency of Q can be given from the axioms of PRA.

[Rupert]

If we assume that I've definitely decided to spend a period of time not applying myself to work, then "nothing else to do" is more or less correct in the sense that my options for alternative ways to occupy myself are fairly limited. But I could have gone for a walk in the park, I guess, or read one of the books in my office. Or written an email to a friend.

And you can of course make the claim that in terms of fulfilling my obligations to my employer and also in terms of the rational pursuit of my own self-interest, it really would be a good idea to continue working even though I feel inclined to take a break. That point of view could definitely be argued.

As to your suggestion that I haven't really hit on what the "true reason" is, I don't know. You could be right, but I don't really know why I would think that. I guess you're trying to say that some aspect of my psychology is involved which I'm somehow averting my eyes from. Or something. As I say, I don't know.

[Rupert]

On Thursday, November 21, 2013 6:19:46 PM UTC+1, Nam Nguyen wrote:
> On 21/11/2013 4:11 AM, Rupert wrote:
> > On Thursday, November 21, 2013 8:23:28 AM UTC+1, Marshall wrote:
> >> So Rupert. Now it is possible to see clearly the sort of person you're arguing
> >> with: someone who dismisses your disproof of his claim as being merely
> >> "some sort of counterexample" and follows it up by exhorting you to
> >> "please remember things straight."
>
> >> Consider how much progress you're going to make discussing things
> >> with someone who sees counterexamples to universal statements
> >> as irrelevant. Now think about how many years you've been arguing
> >> with Nam; how many hours spent at the keyboard.
>
> >> Now ask yourself why.
>
> > Because I'm feeling the need to take a break from reading about semisimple algebraic groups, and there's nothing else to do, and it's mildly diverting.
>
> Freaking pathetic! Liar!
>

What leads you to believe that I'm lying? What do you think my motivation is?

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 21/11/2013 4:35 AM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>

...

>>> Here's the definition of T having M as its model:
>>>
>>> (M is a model of T) df= (M is a structure) and
>>> (T is consistent) and
>>> ((S is a statement-theorem in T) => (M |= S)).
>>
>> That's *not* what everyone else uses as definition of model.
>
> You don't have a proof for that.

It's not in any text I know.

Can you point me to anywhere this definition appears, apart
from your own claim?

> The only thing it matters
> is whether or not it's a technically correct definition,
> and it is.
>
>> Look at Shoenfield, for once.
>
> _You_ should look it at least more twice: his definition
> contains a circularity. And in any rate from there you can
> conclude my there's nothing technically wrong with my definition above.

I have it in front of me.
His definition has no condition that T is consistent.

"By a model of a theory T, we mean a structure for L(T) in which all
the non-logical axioms of T are valid."

The definition of valid is defined purely in terms of structures
(no mention of proof rules).

>> In your terms, here's something nore like the normal definition:
>>
>> (M is a model of T) df= (M is a structure) and
>> ((S is a statement-theorem in T) => (M |= S)).
>>
>>
>>> This definition presupposes T is consistent, because otherwise ~Ax[x=x]
>>> would be true in M, a violation of language structure definition.

Not at all; from the definition, we can work out that ~Ax[x=x] has
no models. And we can deduce that therefore it is inconsistent.
(Take any formula Form in the relevant language.
Any such formula is a logical consequence of ~Ax[x=x], since
it is true in all models of ~Ax[x=x], vacuously (since the set
of models is empty). Completeness tells us that Form is provable.)

> I've addressed the issue with Shoenfield right above. Why don't
> you just review what I've just said and let me know specifically
> what's wrong with my definition of model of a T.
>
> And I mean "specifically", not innuendo, or "blah blah ..." kind
> of arguments.

Specifically, you introduce a condition of consistency which is not
present in Shoenfield.

Generally, a definition of "being a model" has no need to introduce any notion
of proof at all.


--
Alan Smaill

[Peter Percival]

Rupert wrote:

> Well, I can certainly give you a definition, but if you want me to
> give you a citation to Shoenfield's definition

Page 42.

> I'll have to wait
> until I get home, I'm in the office at the moment.

--
Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.

[Marshall]

Narcissism?

[Rupert]

Could be one possible explanation of why he holds the view, I guess.

I mean, he might think there's something excessively inflammatory about the choice of language, maybe, but I don't really think so, I'm just saying that I find the argument to be entirely without merit. I don't know if he would find the phrase "entirely without merit" to be preferable.

I guess we could also bring up the point that he's called me "pathetic" more than once. So I wonder exactly how this fits into his moral views. Maybe he would say it's acceptable when it's retaliation. I have no idea.

[Nam Nguyen]

On 22/11/2013 3:25 AM, Rupert wrote:
> On Friday, November 22, 2013 3:11:06 AM UTC+1, Nam Nguyen wrote:
>> On 21/11/2013 10:41 AM, Rupert wrote:
>>> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
>>>> On 21/11/2013 4:09 AM, Rupert wrote:
>>>>> On Thursday, November 21, 2013 8:23:05 AM UTC+1, Nam Nguyen wrote:
>>
>>>>>> But didn't you say PRA was used there?
>>
>>>>> The argument can be formalised as a formal proof in PRA.
>>
>>>> But is PRA consistent?
>>
>>> Yes.
>>
>> But how would you prove it, using FOL(=) definition of consistency?
>>
>> Or are you just assuming PRA is consistent?
>>
>
> I believe that it's consistent. If you're asking "Can you prove it", then that would obviously depend on which axioms you're going to allow me to use. But that's neither here nor there, really, because if you've got genuine doubts about the consistency of PRA then any proof that I can give you is not going to be epistemologically convincing for you.
>
> Do you, in fact, have genuine doubts about the consistency of PRA? Do you regard it as an open question?

Open question or not, one certainly can prove PRA is inconsistent.

>
> I mean, I suppose I can say that in some sense it's an open question whether I'm currently using a computer to communicate with someone on the internet. But I don't really feel any need to entertain serious doubts about the matter. If you do have serious doubts about the consistency of PRA or even the consistency of Q, then that's fine. However, it remains an objective fact that a proof of the consistency of Q can be given from the axioms of PRA.

As just mentioned, one certainly can prove PRA is inconsistent: so
your belief that PRA is consistent is wrong.

[Nam Nguyen]

On 22/11/2013 3:56 AM, Alan Smaill wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> On 21/11/2013 4:35 AM, Alan Smaill wrote:
>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>
> ...
>>>> Here's the definition of T having M as its model:
>>>>
>>>> (M is a model of T) df= (M is a structure) and
>>>> (T is consistent) and
>>>> ((S is a statement-theorem in T) => (M |= S)).
>>>
>>> That's *not* what everyone else uses as definition of model.
>>
>> You don't have a proof for that.
>
> It's not in any text I know.
>
> Can you point me to anywhere this definition appears, apart
> from your own claim?

I don't have to show you anything here: I was only challenging to prove
your own statement.

>
>> The only thing it matters
>> is whether or not it's a technically correct definition,
>> and it is.
>>
>>> Look at Shoenfield, for once.
>>
>> _You_ should look it at least more twice: his definition
>> contains a circularity. And in any rate from there you can
>> conclude my there's nothing technically wrong with my definition above.
>
> I have it in front of me.
> His definition has no condition that T is consistent.
>
> "By a model of a theory T, we mean a structure for L(T) in which all
> the non-logical axioms of T are valid."
>
> The definition of valid is defined purely in terms of structures
> (no mention of proof rules).

Is that the only sentence he wrote in that _short paragraph_ ?
Would you be able to excerpt that short paragraph here?

>
>>> In your terms, here's something nore like the normal definition:
>>>
>>> (M is a model of T) df= (M is a structure) and
>>> ((S is a statement-theorem in T) => (M |= S)).
>>>
>>>
>>>> This definition presupposes T is consistent, because otherwise ~Ax[x=x]
>>>> would be true in M, a violation of language structure definition.
>
> Not at all; from the definition, we can work out that ~Ax[x=x] has
> no models. And we can deduce that therefore it is inconsistent.
> (Take any formula Form in the relevant language.
> Any such formula is a logical consequence of ~Ax[x=x], since
> it is true in all models of ~Ax[x=x], vacuously (since the set
> of models is empty). Completeness tells us that Form is provable.)
>
>> I've addressed the issue with Shoenfield right above. Why don't
>> you just review what I've just said and let me know specifically
>> what's wrong with my definition of model of a T.
>>
>> And I mean "specifically", not innuendo, or "blah blah ..." kind
>> of arguments.
>
> Specifically, you introduce a condition of consistency which is not
> present in Shoenfield.
>
> Generally, a definition of "being a model" has no need to introduce any notion
> of proof at all.


--

[Rupert]

On Saturday, November 23, 2013 7:22:37 AM UTC+1, Nam Nguyen wrote:
> On 22/11/2013 3:25 AM, Rupert wrote:
> > On Friday, November 22, 2013 3:11:06 AM UTC+1, Nam Nguyen wrote:
> >> On 21/11/2013 10:41 AM, Rupert wrote:
> >>> On Thursday, November 21, 2013 5:04:30 PM UTC+1, Nam Nguyen wrote:
> >>>> On 21/11/2013 4:09 AM, Rupert wrote:
> >>>>> On Thursday, November 21, 2013 8:23:05 AM UTC+1, Nam Nguyen wrote:
>
> >>>>>> But didn't you say PRA was used there?
>
> >>>>> The argument can be formalised as a formal proof in PRA.
>
> >>>> But is PRA consistent?
>
> >>> Yes.
>
> >> But how would you prove it, using FOL(=) definition of consistency?
>
> >> Or are you just assuming PRA is consistent?
>
> > I believe that it's consistent. If you're asking "Can you prove it", then that would obviously depend on which axioms you're going to allow me to use. But that's neither here nor there, really, because if you've got genuine doubts about the consistency of PRA then any proof that I can give you is not going to be epistemologically convincing for you.
>
> > Do you, in fact, have genuine doubts about the consistency of PRA? Do you regard it as an open question?
>
> Open question or not, one certainly can prove PRA is inconsistent.
>

What's the proof?

> > I mean, I suppose I can say that in some sense it's an open question whether I'm currently using a computer to communicate with someone on the internet. But I don't really feel any need to entertain serious doubts about the matter. If you do have serious doubts about the consistency of PRA or even the consistency of Q, then that's fine. However, it remains an objective fact that a proof of the consistency of Q can be given from the axioms of PRA.
>
> As just mentioned, one certainly can prove PRA is inconsistent: so
> your belief that PRA is consistent is wrong.
>

Show me the proof.

[Nam Nguyen]

How about (( PRA + ~x=x) |- ~CON(PRA)) being true?

Wouldn't this constitute a First Order _proof_ of PRA not being
consistent (per your "obviously depend on which axioms")?

So you should not have said "I _believe_ that it's consistent"!
[Emphasis is mine].

What does your "believe" have to do with _rigidity_ of logic and
_definition of consistency_ which _mentions only ONE theory_ here?

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 22/11/2013 3:56 AM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 21/11/2013 4:35 AM, Alan Smaill wrote:
>>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>>
>> ...
>>>>> Here's the definition of T having M as its model:
>>>>>
>>>>> (M is a model of T) df= (M is a structure) and
>>>>> (T is consistent) and
>>>>> ((S is a statement-theorem in T) => (M |= S)).
>>>>
>>>> That's *not* what everyone else uses as definition of model.
>>>
>>> You don't have a proof for that.
>>
>> It's not in any text I know.
>>
>> Can you point me to anywhere this definition appears, apart
>> from your own claim?
>
> I don't have to show you anything here: I was only challenging to prove
> your own statement.

OK, so you do not know of anyone else using your proposed definition.
I take it, therefore, that you do not think that Shoenfield
gives your definition.

>>> The only thing it matters
>>> is whether or not it's a technically correct definition,
>>> and it is.
>>>
>>>> Look at Shoenfield, for once.
>>>
>>> _You_ should look it at least more twice: his definition
>>> contains a circularity. And in any rate from there you can
>>> conclude my there's nothing technically wrong with my definition above.
>>
>> I have it in front of me.
>> His definition has no condition that T is consistent.
>>
>> "By a model of a theory T, we mean a structure for L(T) in which all
>> the non-logical axioms of T are valid."
>>
>> The definition of valid is defined purely in terms of structures
>> (no mention of proof rules).
>
> Is that the only sentence he wrote in that _short paragraph_ ?
> Would you be able to excerpt that short paragraph here?

"By a model of a theory T, we mean a structure for L(T) in which all

the non-logical axioms of T are valid. A formula is valid in T if
it is valid in every model of T; equivalently, if it is a logical
consequence of the nonlogical axioms of T."

So, no reference to any proof system, or to (syntactic) consistency.

>>>> In your terms, here's something nore like the normal definition:
>>>>
>>>> (M is a model of T) df= (M is a structure) and
>>>> ((S is a statement-theorem in T) => (M |= S)).
>>>>
>>>>
>>>>> This definition presupposes T is consistent, because otherwise ~Ax[x=x]
>>>>> would be true in M, a violation of language structure definition.
>>
>> Not at all; from the definition, we can work out that ~Ax[x=x] has
>> no models. And we can deduce that therefore it is inconsistent.
>> (Take any formula Form in the relevant language.
>> Any such formula is a logical consequence of ~Ax[x=x], since
>> it is true in all models of ~Ax[x=x], vacuously (since the set
>> of models is empty). Completeness tells us that Form is provable.)
>>
>>> I've addressed the issue with Shoenfield right above. Why don't
>>> you just review what I've just said and let me know specifically
>>> what's wrong with my definition of model of a T.
>>>
>>> And I mean "specifically", not innuendo, or "blah blah ..." kind
>>> of arguments.
>>
>> Specifically, you introduce a condition of consistency which is not
>> present in Shoenfield.
>>
>> Generally, a definition of "being a model" has no need to introduce
>> any notion of proof at all.

--

Alan Smaill

[Nam Nguyen]

On 23/11/2013 8:50 AM, Alan Smaill wrote:
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> On 22/11/2013 3:56 AM, Alan Smaill wrote:
>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>
>>>> On 21/11/2013 4:35 AM, Alan Smaill wrote:
>>>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>>>
>>> ...
>>>>>> Here's the definition of T having M as its model:
>>>>>>
>>>>>> (M is a model of T) df= (M is a structure) and
>>>>>> (T is consistent) and
>>>>>> ((S is a statement-theorem in T) => (M |= S)).
>>>>>
>>>>> That's *not* what everyone else uses as definition of model.
>>>>
>>>> You don't have a proof for that.
>>>
>>> It's not in any text I know.
>>>
>>> Can you point me to anywhere this definition appears, apart
>>> from your own claim?
>>
>> I don't have to show you anything here: I was only challenging to prove
>> your own statement.
>
> OK, so you do not know of anyone else using your proposed definition.
> I take it, therefore, that you do not think that Shoenfield
> gives your definition.

Word to word, No. But then there's such a thing as equivalence of
_different_ definitions. Isosceles triangle can be defined in term
of "side" or "angle", correct?

More about Shoenfield's definition of model of a T below.

>
>>>> The only thing it matters
>>>> is whether or not it's a technically correct definition,
>>>> and it is.
>>>>
>>>>> Look at Shoenfield, for once.
>>>>
>>>> _You_ should look it at least more twice: his definition
>>>> contains a circularity. And in any rate from there you can
>>>> conclude my there's nothing technically wrong with my definition above.
>>>
>>> I have it in front of me.
>>> His definition has no condition that T is consistent.
>>>
>>> "By a model of a theory T, we mean a structure for L(T) in which all
>>> the non-logical axioms of T are valid."
>>>
>>> The definition of valid is defined purely in terms of structures
>>> (no mention of proof rules).
>>
>> Is that the only sentence he wrote in that _short paragraph_ ?
>> Would you be able to excerpt that short paragraph here?
>
> "By a model of a theory T, we mean a structure for L(T) in which all
> the non-logical axioms of T are valid. A formula is valid in T if
> it is valid in every model of T; equivalently, if it is a logical
> consequence of the nonlogical axioms of T."

Before examining the usefulness, validity, of a _circular definition_
here, do you concede that his definition of a model of a theory T is
circular, as I've just pointed out to you above (my "his definition
contains a circularity")?

>
> So, no reference to any proof system, or to (syntactic) consistency.

"So, no reference to any proof system"? I was about to ask what then
you'd think his "logical consequence of the nonlogical axioms of T"
means, but how about your conceding about my circularity-request
above first.

>>>>> In your terms, here's something nore like the normal definition:
>>>>>
>>>>> (M is a model of T) df= (M is a structure) and
>>>>> ((S is a statement-theorem in T) => (M |= S)).
>>>>>
>>>>>
>>>>>> This definition presupposes T is consistent, because otherwise ~Ax[x=x]
>>>>>> would be true in M, a violation of language structure definition.
>>>
>>> Not at all; from the definition, we can work out that ~Ax[x=x] has
>>> no models. And we can deduce that therefore it is inconsistent.
>>> (Take any formula Form in the relevant language.
>>> Any such formula is a logical consequence of ~Ax[x=x], since
>>> it is true in all models of ~Ax[x=x], vacuously (since the set
>>> of models is empty). Completeness tells us that Form is provable.)
>>>
>>>> I've addressed the issue with Shoenfield right above. Why don't
>>>> you just review what I've just said and let me know specifically
>>>> what's wrong with my definition of model of a T.
>>>>
>>>> And I mean "specifically", not innuendo, or "blah blah ..." kind
>>>> of arguments.
>>>
>>> Specifically, you introduce a condition of consistency which is not
>>> present in Shoenfield.
>>>
>>> Generally, a definition of "being a model" has no need to introduce
>>> any notion of proof at all.


--

[Jim Burns]

On 11/23/2013 9:58 AM, Nam Nguyen wrote:
> On 23/11/2013 12:26 AM, Rupert wrote:
>> On Saturday, November 23, 2013 7:22:37 AM UTC+1,
>> Nam Nguyen wrote:

>>> As just mentioned, one certainly can prove PRA is inconsistent:
>>> so your belief that PRA is consistent is wrong.
>>
>> Show me the proof.
>
> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>
> Wouldn't this constitute a First Order _proof_ of PRA not being
> consistent (per your "obviously depend on which axioms")?

Remember what you have most recently claimed: That Rupert is
wrong.

If you take as your axioms an inconsistent set of statements,
such as your ( PRA + ~x=x) , then _every_ statement follows
from that, ~CON(PRA) but CON(PRA) too. (I'm sure you're aware
of that, which is why you did not bother to prove the above.)

If you were to argue in good faith (whatever that means
when the assumptions are intentionally inconsistent),
then you would not be able to assert that anything is
wrong -- or, rather, that a particular statement is wrong,
but you would have to agree that it is right too ...
or something like that.

Just to be clear here: You agree that Rupert is correct.
True? Keep in mind that (( PRA + ~x=x) |- CON(PRA)).

[Nam Nguyen]

On 23/11/2013 9:41 AM, Jim Burns wrote:
> On 11/23/2013 9:58 AM, Nam Nguyen wrote:
>> On 23/11/2013 12:26 AM, Rupert wrote:
>>> On Saturday, November 23, 2013 7:22:37 AM UTC+1,
>>> Nam Nguyen wrote:
>
>>>> As just mentioned, one certainly can prove PRA is inconsistent:
>>>> so your belief that PRA is consistent is wrong.
>>>
>>> Show me the proof.
>>
>> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>>
>> Wouldn't this constitute a First Order _proof_ of PRA not being
>> consistent (per your "obviously depend on which axioms")?
>
> Remember what you have most recently claimed: That Rupert is
> wrong.
>
> If you take as your axioms an inconsistent set of statements,
> such as your ( PRA + ~x=x) , then _every_ statement follows
> from that, ~CON(PRA) but CON(PRA) too. (I'm sure you're aware
> of that, which is why you did not bother to prove the above.)
>
> If you were to argue in good faith (whatever that means
> when the assumptions are intentionally inconsistent),

Try not to conveniently freaking-snip relevant part of the argument
between him and me and imply I'm not arguing in good faith.

I'm very well aware an inconsistent theory would prove everything
here and my using (( PRA + ~x=x) |- ~CON(PRA)) only serves
to a counter point to his "believe" that PRA is consistent,
_without regard to standard definition of consistency_ .
(Note also his "obviously depend on which axioms").

Before pouring acidity into a technical debate with your
ill-intention, why don't you examine the technical points
in the debate more carefully!

> then you would not be able to assert that anything is
> wrong -- or, rather, that a particular statement is wrong,
> but you would have to agree that it is right too ...
> or something like that.
>
> Just to be clear here: You agree that Rupert is correct.
> True? Keep in mind that (( PRA + ~x=x) |- CON(PRA)).

No. I've argued that _his counter argument_ to my meta
theorem about it being impossible to assert consistency
_is wrong_ .

Are you now clear on what is the key point being debated here?

[Jim Burns]

On 11/23/2013 11:57 AM, Nam Nguyen wrote:
> On 23/11/2013 9:41 AM, Jim Burns wrote:
>> On 11/23/2013 9:58 AM, Nam Nguyen wrote:
>>> On 23/11/2013 12:26 AM, Rupert wrote:
>>>> On Saturday, November 23, 2013 7:22:37 AM UTC+1,
>>>> Nam Nguyen wrote:

>>>>> As just mentioned, one certainly can prove PRA is inconsistent:
>>>>> so your belief that PRA is consistent is wrong.
>>>>
>>>> Show me the proof.
>>>
>>> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>>>
>>> Wouldn't this constitute a First Order _proof_ of PRA not being
>>> consistent (per your "obviously depend on which axioms")?
>>
>> Remember what you have most recently claimed: That Rupert is
>> wrong.
>>
>> If you take as your axioms an inconsistent set of statements,
>> such as your ( PRA + ~x=x) , then _every_ statement follows
>> from that, ~CON(PRA) but CON(PRA) too. (I'm sure you're aware
>> of that, which is why you did not bother to prove the above.)
>>
>> If you were to argue in good faith (whatever that means
>> when the assumptions are intentionally inconsistent),
>
> Try not to conveniently freaking-snip relevant part of the argument
> between him and me and imply I'm not arguing in good faith.

Assuming, for the sake of argument, that you assert that
you *are* arguing in good faith, what would you mean by
"arguing in good faith"? (FYI, when I use that phrase, I mean
it in the most common way it is used.)

I ask because you have been using various words and phrases
in your own, personal way. It is probably not a coincidence that
it seems at first glance as though you might be making good arguments
for whatever your point-of-the-moment is. It is only when one finds out
what you personally mean by a critical phrase that the gaping holes
in your argument become obvious.

This present point-of-the-moment serves as a good example, but
I have others, if you don't like this one. To review:

-- Rupert said he believes Con(RPA).

-- You presented an axiom set that proves ~Con(RPA) and said
that this shows he was wrong.

-- With nearly all commonly seen axiom sets, this would have been
enough to show that Rupert was wrong, at least with regard to
that axiom set, but nearly all axiom sets are are consistent
or at the very least not obviously inconsistent. With a consistent
axiom set, proving ~Con(PRA) would have been enough to show
that one could *not* prove Con(RPA). That would have shown that Rupert
would not have been justified in believing Con(RPA),
when assuming your axioms.

-- However, your axiom set was inconsistent and obviously so.
Yes, it proves ~Con(RPA), but it does not show that Rupert
was *not* justified in believing Con(RPA), because it also
proves Con(RPA), justifying Rupert -- at least, justifying him
to exactly the same extent that you regard ~Con(RP) as
justified, whatever that may be.

You presented an argument that makes it *sound as though*
Rupert was not justified in believing Con(RPA), but did not.
In order to do this, you used a feature most likely covered
in every introductory logic course in existence. The possible
explanations for this seem to boil down to (1) you are
profoundly ignorant, or (2) you were not arguing in good faith
(as I use the phrase).

You insist below that you were not ignorant, and this once
I believe you.

So: what do *you* mean by "not arguing in good faith"?

> I'm very well aware an inconsistent theory would prove everything
> here and my using (( PRA + ~x=x) |- ~CON(PRA)) only serves
> to a counter point to his "believe" that PRA is consistent,
> _without regard to standard definition of consistency_ .
> (Note also his "obviously depend on which axioms").
>
> Before pouring acidity into a technical debate with your
> ill-intention, why don't you examine the technical points
> in the debate more carefully!
>
>> then you would not be able to assert that anything is
>> wrong -- or, rather, that a particular statement is wrong,
>> but you would have to agree that it is right too ...
>> or something like that.
>>
>> Just to be clear here: You agree that Rupert is correct.
>> True? Keep in mind that (( PRA + ~x=x) |- CON(PRA)).
>
> No. I've argued that _his counter argument_ to my meta
> theorem about it being impossible to assert consistency
> _is wrong_ .
>
> Are you now clear on what is the key point being debated here?
>

Do you agree that (( PRA + ~x=x) |- CON(PRA))?

If consistency follows logically from your axioms, what does the
phrase "impossible to assert consistency" mean when you use it?

Please speculate on why logicians and mathematicians almost always
avoid using inconsistent axioms. Thanks in advance.

[Peter Percival]

Jim Burns wrote:

> [Excellent stuff snipped]

> The possible
> explanations for this seem to boil down to (1) you are
> profoundly ignorant, or (2) you were not arguing in good faith
> (as I use the phrase).

Or perhaps both. It is my belief that Nam is mentally defective (for
which reason I have given up responding to his posts). Yes, he does
argue in bad faith but in such a crass manner that he must be mentally
defective if he thinks anyone is taken in by it, or that it serves any
purpose at all. His profound ignorance is just a by-product of his
mental deficiency.

I am slightly puzzled that a mentally defective person should possess
Shoenfield's text book, which is rather advanced; and he once said he
had possessed (and has since disposed of) Kleene's /Introduction to
metamathematics/, also an advanced work. But it is only a slight
puzzlement; a plausible explanation comes to mind: he inherited both
books from a relative. Another possibility is that he was once
reasonably intelligent (I think he once told Moe Blee that he had a
degree in mathematics) and acquired both books in order to study their
subject matter, but subsequently suffered brain damage as the result of
an accident or illness.

[Nam Nguyen]

(Make it 2 of us: same here).

>
> I ask because you have been using various words and phrases
> in your own, personal way.

That's what you _personally_ think.

> It is probably not a coincidence that
> it seems at first glance as though you might be making good arguments
> for whatever your point-of-the-moment is.

Yes. Good arguments tend to start out with a good "first glance".

> It is only when one finds out
> what you personally mean by a critical phrase that the gaping holes
> in your argument become obvious.

Idiotic ranting. You and Rupert _repeatedly ignored_ the critical
phrase (of this whole argument) which is that the standard definition
of consistency references only to _ONE_ theory: the underlying one.

Hence to claim Q is consistent based on another theory (PRA), as Rupert
has claimed, is invalid by FOL(=) definition of consistency.

_When will you and Rupert acknowlegde this fact_ ?

>
> This present point-of-the-moment serves as a good example, but
> I have others, if you don't like this one. To review:
>
> -- Rupert said he believes Con(RPA).
>
> -- You presented an axiom set that proves ~Con(RPA) and said
> that this shows he was wrong.

Right. If you know _why_ I said he was wrong there, of course.

>
> -- With nearly all commonly seen axiom sets, this would have been
> enough to show that Rupert was wrong, at least with regard to
> that axiom set, but nearly all axiom sets are are consistent
> or at the very least not obviously inconsistent.

But have Rupert and Jim proven these axiom sets are consistent,
_using the standard consistency-definition that would reference_
_only ONE theory_ ?

> With a consistent
> axiom set, proving ~Con(PRA) would have been enough to show
> that one could *not* prove Con(RPA).

But have Rupert and Jim proven this "consistent" axiom set is
consistent, _using the standard consistency-definition that_
_would reference only ONE theory_ ?

> That would have shown that Rupert
> would not have been justified in believing Con(RPA),
> when assuming your axioms.

Did Rupert prove PRA is consistent, _using the standard consistency_
_definition that would reference only ONE theory_ , or did he just
justify this purported consistency merely by his "believe"?

>
> -- However, your axiom set was inconsistent and obviously so.
> Yes, it proves ~Con(RPA), but it does not show that Rupert
> was *not* justified in believing Con(RPA), because it also
> proves Con(RPA), justifying Rupert -- at least, justifying him
> to exactly the same extent that you regard ~Con(RP) as
> justified, whatever that may be.

You just don't have any clue in what you're talking about.
That ~Con(RPA) is provable here means it's logically justified:
by definition of FOL theorem, and Rupert's claim PRA is consistent
is _not_ logically justified: "believe" is not a valid justification
here.

>
> You presented an argument that makes it *sound as though*
> Rupert was not justified in believing Con(RPA), but did not.
> In order to do this, you used a feature most likely covered
> in every introductory logic course in existence. The possible
> explanations for this seem to boil down to (1) you are
> profoundly ignorant, or (2) you were not arguing in good faith
> (as I use the phrase).

Go and fuck yourself, Jim Burns.

My argument here with (( PRA + ~x=x) |- ~CON(PRA)) being true
serves a notice to Rupert his "believing" that PRA is consistent
is _NOT a VALID argument_ in this context: "believe" doesn't
have a FOL definition and argument based on outside FOL definitions
are _NOT_ valid!

If you jump into a conversation but _not recognize the contexts_
where certain things were said, then STFU, instead of accusing
people (1) or (2)!

>
> You insist below that you were not ignorant, and this once
> I believe you.
>
> So: what do *you* mean by "not arguing in good faith"?
>
>> I'm very well aware an inconsistent theory would prove everything
>> here and my using (( PRA + ~x=x) |- ~CON(PRA)) only serves
>> to a counter point to his "believe" that PRA is consistent,
>> _without regard to standard definition of consistency_ .
>> (Note also his "obviously depend on which axioms").
>>
>> Before pouring acidity into a technical debate with your
>> ill-intention, why don't you examine the technical points
>> in the debate more carefully!
>>
>>> then you would not be able to assert that anything is
>>> wrong -- or, rather, that a particular statement is wrong,
>>> but you would have to agree that it is right too ...
>>> or something like that.
>>>
>>> Just to be clear here: You agree that Rupert is correct.
>>> True? Keep in mind that (( PRA + ~x=x) |- CON(PRA)).
>>
>> No. I've argued that _his counter argument_ to my meta
>> theorem about it being impossible to assert consistency
>> _is wrong_ .
>>
>> Are you now clear on what is the key point being debated here?
>>

Why didn't you answer this question of mine?

>
> Do you agree that (( PRA + ~x=x) |- CON(PRA))?
>
> If consistency follows logically from your axioms, what does the
> phrase "impossible to assert consistency" mean when you use it?
>
> Please speculate on why logicians and mathematicians almost always
> avoid using inconsistent axioms. Thanks in advance.

[Nam Nguyen]

On 23/11/2013 2:21 PM, Peter Percival wrote:
> Jim Burns wrote:
>
>> [Excellent stuff snipped]
>
>> The possible
>> explanations for this seem to boil down to (1) you are
>> profoundly ignorant, or (2) you were not arguing in good faith
>> (as I use the phrase).
>
> Or perhaps both. It is my belief that Nam is mentally defective (for
> which reason I have given up responding to his posts).

Peter Percival is a liar.

[Rupert]

On Saturday, November 23, 2013 3:58:45 PM UTC+1, Nam Nguyen wrote:
>
> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>
> Wouldn't this constitute a First Order _proof_ of PRA not being
> consistent (per your "obviously depend on which axioms")?
>
> So you should not have said "I _believe_ that it's consistent"!
> [Emphasis is mine].
>
> What does your "believe" have to do with _rigidity_ of logic and
> _definition of consistency_ which _mentions only ONE theory_ here?
>

The fact that you can prove anything in an inconsistent theory obviously has no bearing on the question of whether PRA is consistent.

If you make a claim that you can prove PRA is inconsistent, then it's reasonable to assume that you mean that you can derive a contradiction from the axioms of PRA.

It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.

[Nam Nguyen]

On 23/11/2013 3:04 PM, Rupert wrote:
> On Saturday, November 23, 2013 3:58:45 PM UTC+1, Nam Nguyen wrote:
>>
>> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>>
>> Wouldn't this constitute a First Order _proof_ of PRA not being
>> consistent (per your "obviously depend on which axioms")?
>>
>> So you should not have said "I _believe_ that it's consistent"!
>> [Emphasis is mine].
>>
>> What does your "believe" have to do with _rigidity_ of logic and
>> _definition of consistency_ which _mentions only ONE theory_ here?
>>
>
> The fact that you can prove anything in an inconsistent theory obviously has no bearing on the question of whether PRA is consistent.

Right. And that's my point which Jim Burns apparently didn't recognize.

But by the same token, the fact Rupert _believes_ PRA is consistent also

obviously has no bearing on the question of whether PRA is consistent.

>
> If you make a claim that you can prove PRA is inconsistent, then it's reasonable to assume that you mean that you can derive a contradiction from the axioms of PRA.

But see: you missed my point. I claimed _here_ PRA is inconsistent based
on two invalid assumption:

1) Proving PRA's (in)consistency _using another theory_ is invalid.

2) Proving PRA's (in)consistency _using extra definitions_ is invalid:
here CON(PRA) and ~CON(PRA) are used as definitions of consistency
and inconsistency, which are outside (hence extra) FOL(=) definition
of consistency and inconsistency, hence arguments using them are
invalid.

>
> It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.
>

I'm sorry Rupert that I don't know how else I could convey to you
that you're simply wrong here: logically, one's own opinion wouldn't
count.

What is expected is meta level proof of consistency [based on FOL(=)
definition of consistency] which no one can produce: FOL(=) definition
of consistency WILL NOT grant one a way to prove such consistency,
_unlike the case of inconsistency_ .

Why, Rupert, couldn't you understand/acknowledge such a simple fact?

[Jim Burns]

[Note: If I respond again to this sub-thread,
I'm going to start hacking away useless text.]

On occasion, you've been asked to provide examples of
*anyone else* using them your way. Recently, for example,
Rupert asked you for examples of "consistent" used the
way you wanted to use it to make your point-of-the-moment
about consistency. You didn't give any. (You still can,
if you've got any. Do you?)

The various words and phrases of which I am thinking
are understood by others the way I use them, in fact, by
you too on occasions other than when those "controversial"
definitions are being used by you to make a point.
Your possibly infinite 0 comes to mind as an example.

So, no. It's not merely what I _personally_ think.

>> It is probably not a coincidence that
>> it seems at first glance as though you might be making good arguments
>> for whatever your point-of-the-moment is.
>
> Yes. Good arguments tend to start out with a good "first glance".
>
>> It is only when one finds out
>> what you personally mean by a critical phrase that the gaping holes
>> in your argument become obvious.
>
> Idiotic ranting. You and Rupert _repeatedly ignored_ the critical
> phrase (of this whole argument) which is that the standard definition
> of consistency references only to _ONE_ theory: the underlying one.
>
> Hence to claim Q is consistent based on another theory (PRA),
> as Rupert has claimed, is invalid by FOL(=) definition of
> consistency.

Please provide a reference to someone other than yourself using
this non-personal definition of consistency you want to use.

(i) "Believe" was not the whole of Rupert's justification.

(ii) Con(RPA) is "logically justified" to exactly the same degree
as ~Con(RPA), and I mean *exactly* , the same proof, just swapping
"Con(RPA)" for "~Con(RPA)". ~Con(RPA) is also justified to *exactly*
the same degree as "Nam Nguyen is a bowl of petunias currently thinking
'Oh no, not again'".

The way you are using "logically justified" does not seem very useful.
Maybe it is "logically justified" a la Nam. Why should anyone care,
yourself included? What color are your petals, Nam?

I demonstrated that I understood your key point by demolishing
that point -- assuming that you are not using your own,
personal meaning of "impossible to assert consistency", which
you say you are not doing.

>> Do you agree that (( PRA + ~x=x) |- CON(PRA))?
>>
>> If consistency follows logically from your axioms, what does the
>> phrase "impossible to assert consistency" mean when you use it?
>>
>> Please speculate on why logicians and mathematicians almost always
>> avoid using inconsistent axioms. Thanks in advance.

Your turn: Please speculate as previously requested.

[George Greene]

On Saturday, November 23, 2013 1:22:37 AM UTC-5, Nam Nguyen wrote:
> Open question or not, one certainly can prove PRA is inconsistent.

Unfortunately for you, however, that "one" IS NOT *you*.

[Nam Nguyen]

I'm a little bit sick and tired of Inquisition style of responding
to my post, so why don't _you_ re-read my post and understand why
I said what I said.

Just freaking sniping my entire post so you could utter 'that "one" IS
NOT *you*' is saying jack, besides being pathetic!

I'm amazing that those who thought and/or claimed they'd defend the
traditional mathematics and logic would have an attitude of street
gangsters and mobsters, which is rather sad news for the venerable
tradition of rigidity in mathematical reasoning.

[Of course I'm not saying all posters/viewers in sci.logic would take
mathematical logic for grant, as you and few other opponents of mine
have].

[Nam Nguyen]

On 23/11/2013 11:38 PM, Nam Nguyen wrote:
> On 23/11/2013 9:06 PM, George Greene wrote:
>> On Saturday, November 23, 2013 1:22:37 AM UTC-5, Nam Nguyen wrote:
>>> Open question or not, one certainly can prove PRA is inconsistent.
>>
>> Unfortunately for you, however, that "one" IS NOT *you*.
>
> I'm a little bit sick and tired of Inquisition style of responding
> to my post, so why don't _you_ re-read my post and understand why
> I said what I said.
>
> Just freaking sniping my entire post so you could utter 'that "one" IS
> NOT *you*' is saying jack, besides being pathetic!

Seriously, George, you're here not better at all compared to the crank
you've on and off argued with: your reasoning and that of those crank
are of the same pathetic style!

[Jim Burns]

On 11/23/2013 5:25 PM, Nam Nguyen wrote:
> On 23/11/2013 3:04 PM, Rupert wrote:
>> On Saturday, November 23, 2013 3:58:45 PM UTC+1, Nam Nguyen wrote:

>>> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>>>
>>> Wouldn't this constitute a First Order _proof_ of PRA not being
>>> consistent (per your "obviously depend on which axioms")?
>>>
>>> So you should not have said "I _believe_ that it's consistent"!
>>> [Emphasis is mine].
>>>
>>> What does your "believe" have to do with _rigidity_ of logic and
>>> _definition of consistency_ which _mentions only ONE theory_ here?
>>
>> The fact that you can prove anything in an inconsistent theory
>> obviously has no bearing on the question of whether PRA is consistent.
>
> Right. And that's my point which Jim Burns apparently didn't recognize.

For the record, I am willing to admit that the fact that you can
prove anything in an inconsistent theory has no bearing on the

question of whether PRA is consistent.

I will go even further: It would have been breathtakingly stupid for
anyone to introduce an inconsistent theory and treat the consequences
as though they had any value -- either breathtakingly stupid not
to realize that it had no value or breathtakingly stupid not to
realize everyone else knows it had no value.

I'm sure Nam agrees with me, as that is the point he apparently has
been making to me. Isn't that right, Nam?

[George Greene]

On Tuesday, November 19, 2013 9:37:42 PM UTC-5, Nam Nguyen wrote:
> >> Except that there's a property, namely its truth value, of the
>
> >> finite, syntactical, formula cGC

ForGET "cGC"!! That is EXTRA baggage!
What about*GC* itSELF?? We don't know the truth-value
OF THAT,EITHER!

> >> that we've never known, and
> >> logically there's a chance it's impossible to know.

NO, THERE ISN'T.
If GC is false then it is absolutely certain that it is possible to
know that. The counter-example is finite so it can eventually be found.
You just start with the highest-number-so-far known to satisfy GC
AND KEEP TRYING HIGHER NUMBERS.

If GC is true then it may be harder to confirm that (it may,
for example, be independent of PA and of ZFC) but primality is
not THAT complicated a concept and if this regularity is necessary
"all the way out" then THERE IS an underlying principle causing that,
and all you have to do to "know" that GC is true is identify this
principle. Moreover, there is a very trivial sense in which we
could "know" it -- we could just STIPULATE it. We are treating
things about N as "known" when they are PROVED. You can make ANYthing
provable JUST by making it AN AXIOM. This is a SAFE practice because
IF you are wrong, YOU WILL find that out, SOON enough. In a better
possible world, your personal penance for this claim would be to get
sentenced to deriving new&interesting theorems *from* GC (as an axiom).

[Rupert]

On Saturday, November 23, 2013 11:25:35 PM UTC+1, Nam Nguyen wrote:
> On 23/11/2013 3:04 PM, Rupert wrote:
> > On Saturday, November 23, 2013 3:58:45 PM UTC+1, Nam Nguyen wrote:
>
> >> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>
> >> Wouldn't this constitute a First Order _proof_ of PRA not being
> >> consistent (per your "obviously depend on which axioms")?
>
> >> So you should not have said "I _believe_ that it's consistent"!
>
> >> [Emphasis is mine].
>
> >> What does your "believe" have to do with _rigidity_ of logic and
> >> _definition of consistency_ which _mentions only ONE theory_ here?
>
> > The fact that you can prove anything in an inconsistent theory obviously has no bearing on the question of whether PRA is consistent.
>
> Right. And that's my point which Jim Burns apparently didn't recognize.
>
> But by the same token, the fact Rupert _believes_ PRA is consistent also
> obviously has no bearing on the question of whether PRA is consistent.
>

What's your opinion about the matter? You think it's not possible to know?

> > If you make a claim that you can prove PRA is inconsistent, then it's reasonable to assume that you mean that you can derive a contradiction from the axioms of PRA.
>
> But see: you missed my point. I claimed _here_ PRA is inconsistent based
> on two invalid assumption:
>
> 1) Proving PRA's (in)consistency _using another theory_ is invalid.
>
> 2) Proving PRA's (in)consistency _using extra definitions_ is invalid:
>
> here CON(PRA) and ~CON(PRA) are used as definitions of consistency
> and inconsistency, which are outside (hence extra) FOL(=) definition
> of consistency and inconsistency, hence arguments using them are
> invalid.
>

If you gave a proof of ¬Con(PRA) from axioms which we generally recognised to be such that there was pretty compelling reason for believing them to be true, then that would be a proof that would carry some weight.

> > It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.
>
> I'm sorry Rupert that I don't know how else I could convey to you
> that you're simply wrong here: logically, one's own opinion wouldn't
> count.
>

You're saying I'm wrong. Which bit was wrong? Was it in some way unreasonable for me to express my opinion that PRA is consistent? Have you in fact refuted it?

> What is expected is meta level proof of consistency [based on FOL(=)
> definition of consistency] which no one can produce: FOL(=) definition
> of consistency WILL NOT grant one a way to prove such consistency,
>

I don't know why you think this.

> _unlike the case of inconsistency_ .
>
> Why, Rupert, couldn't you understand/acknowledge such a simple fact?
>

I suppose it must be something to do with my cognitive limitations.

[Nam Nguyen]

On 24/11/2013 7:01 AM, Jim Burns wrote:
> On 11/23/2013 5:25 PM, Nam Nguyen wrote:
>> On 23/11/2013 3:04 PM, Rupert wrote:
>>> On Saturday, November 23, 2013 3:58:45 PM UTC+1, Nam Nguyen wrote:
>
>>>> How about (( PRA + ~x=x) |- ~CON(PRA)) being true?
>>>>
>>>> Wouldn't this constitute a First Order _proof_ of PRA not being
>>>> consistent (per your "obviously depend on which axioms")?
>>>>
>>>> So you should not have said "I _believe_ that it's consistent"!
>>>> [Emphasis is mine].
>>>>
>>>> What does your "believe" have to do with _rigidity_ of logic and
>>>> _definition of consistency_ which _mentions only ONE theory_ here?
>>>
>>> The fact that you can prove anything in an inconsistent theory
>>> obviously has no bearing on the question of whether PRA is consistent.
>>
>> Right. And that's my point which Jim Burns apparently didn't recognize.
>
> For the record, I am willing to admit that the fact that you can
> prove anything in an inconsistent theory has no bearing on the
> question of whether PRA is consistent.

We're in agreement.

>
> I will go even further: It would have been breathtakingly stupid for
> anyone to introduce an inconsistent theory and treat the consequences
> as though they had any value -- either breathtakingly stupid not
> to realize that it had no value or breathtakingly stupid not to
> realize everyone else knows it had no value.

We're in agreement. It'd be breathtakingly stupid for one, in the
course of doing ordinary mathematics, to use the fact that inconsistency
would prove anything,

But, please, we're _NOT_ (for the millionth time) addressing an ordinary
mathematics issue: we're addressing the issue of _validity_ of making
certain meta statements using valid definitions and valid/permissible
rules of reasoning with FOL(=) framework.

The problem here (mostly due to communication) is you (your side) seem
to loose focus that this is a foundation issue _NOT_ an ordinary
mathematics issue, where everything would count, including the fact
that one not having found a proof of inconsistency does _NOT_ logically
eliminate the real possibility it might still inconsistent!

>
> I'm sure Nam agrees with me, as that is the point he apparently has
> been making to me. Isn't that right, Nam?

Again, you've read too much in my little example of inconsistency, and
yet in the end not read enough on the point that it'd serve.

Yes, you would be able to easily find the inconsistency proof in
T = PRA + {~x=x} as being presented! But anyone with a little knowledge
of formalism and programming language can obfuscate ~x=x with an
equivalent set of formulas, long enough, numerous enough (say a trillion
of formulas), obfuscated enough, that the resultant T' equivalent to
T would be so hard to recognize if it's indeed equivalent to the
inconsistent T, hence the question whether or not T' is consistent is
a _real_ question!

Remember the Quinne's ML theory: he obviously first _believed_ it being
consistent, but it turned out to be inconsistent, and equivalent to
ML + (~x=x}!

The point being that my simple inconsistency example serves the notice
that that, say, Rupert's not having found an inconsistency proof in PRA
is not a logical cause to celebrate, to _believe_ PRA is consistent.

Do you now understand my point here regarding to my little
inconsistency example?

[Nam Nguyen]

On 24/11/2013 7:10 AM, George Greene wrote:
> On Tuesday, November 19, 2013 9:37:42 PM UTC-5, Nam Nguyen wrote:
>>>> Except that there's a property, namely its truth value, of the
>>
>>>> finite, syntactical, formula cGC
>
> ForGET "cGC"!! That is EXTRA baggage!

Ok. GC is _not_ equivalent to cGC and Nam wants to discuss
about cGC but George says No to being interested about cGC!

Why then should Nam be interested in George's posting about GC below?

Tell you what, whatever the issue with GC that you feel interested,
why don't you open a new thread and invite people in? Perhaps there
might be some who might be interested in the subject you'd raise there.

But the interest here is _NOT_ about GC.

The point being, George, the 2 formulas are NOT of the same class:
while there's logical ground to believe the truth value of GC be found
(when, say, it's false), there's not even a slightest logical ground
at least to date that you could attempt to defend that ground for such
truth value of cGC, whatever the truth value you'd happen to choose for
your attempt!

Think about that carefully.

> What about*GC* itSELF?? We don't know the truth-value
> OF THAT,EITHER!
>
>>>> that we've never known, and
>>>> logically there's a chance it's impossible to know.
>
> NO, THERE ISN'T.
> If GC is false then it is absolutely certain that it is possible to
> know that. The counter-example is finite so it can eventually be found.
> You just start with the highest-number-so-far known to satisfy GC
> AND KEEP TRYING HIGHER NUMBERS.
>
> If GC is true then it may be harder to confirm that (it may,
> for example, be independent of PA and of ZFC) but primality is
> not THAT complicated a concept and if this regularity is necessary
> "all the way out" then THERE IS an underlying principle causing that,
> and all you have to do to "know" that GC is true is identify this
> principle. Moreover, there is a very trivial sense in which we
> could "know" it -- we could just STIPULATE it. We are treating
> things about N as "known" when they are PROVED. You can make ANYthing
> provable JUST by making it AN AXIOM. This is a SAFE practice because
> IF you are wrong, YOU WILL find that out, SOON enough. In a better
> possible world, your personal penance for this claim would be to get
> sentenced to deriving new&interesting theorems *from* GC (as an axiom).
>

[Nam Nguyen]

But what is the formal definition of "pretty compelling" and "believe"
within FOL(=) framework? Isn't it true it's "pretty compelling" to
someone here in sci.logic to "believe" a number greater than 10^500
_is_ an infinite number?

>
>>> It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.
>>
>> I'm sorry Rupert that I don't know how else I could convey to you
>> that you're simply wrong here: logically, one's own opinion wouldn't
>> count.
>>
>
> You're saying I'm wrong. Which bit was wrong? Was it in some way unreasonable for me to express my opinion that PRA is consistent? Have you in fact refuted it?

Please read my statement carefully. I didn't say it's unreasonable for
anyone to express an opinion! What I said is "logically, one's own
opinion wouldn't", and that's not the same.

>
>> What is expected is meta level proof of consistency [based on FOL(=)
>> definition of consistency] which no one can produce: FOL(=) definition
>> of consistency WILL NOT grant one a way to prove such consistency,
>>
>
> I don't know why you think this.

Because I've already explained it:

>> FOL(=) definition of consistency WILL NOT grant one a way to prove
>> such consistency,

There's so much one can explain, you know.

>
>> _unlike the case of inconsistency_ .
>>
>> Why, Rupert, couldn't you understand/acknowledge such a simple fact?
>>
>
> I suppose it must be something to do with my cognitive limitations.

Tell you what. Can you prove ZFC is consistent, _using only_ its
axioms, FOL(=) rules of inference, and _NOT using_ the concept
of recursion, truths about the natural numbers, language-structure
theoretical truth?

Yes or No please.

[Rupert]

> > If you gave a proof of �Con(PRA) from axioms which we generally recognised to be such that there was pretty compelling reason for believing them to be true, then that would be a proof that would carry some weight.

>
> But what is the formal definition of "pretty compelling" and "believe"
> within FOL(=) framework? Isn't it true it's "pretty compelling" to
> someone here in sci.logic to "believe" a number greater than 10^500
> _is_ an infinite number?
>

The concepts don't have formal definitions. If someone believes that there is pretty compelling reason to believe that 10^500+1 is infinite, then they're wrong. We're not claiming that our beliefs about such matters are infallible.

> >>> It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.
>
> >> I'm sorry Rupert that I don't know how else I could convey to you
> >> that you're simply wrong here: logically, one's own opinion wouldn't
> >> count.
>
> > You're saying I'm wrong. Which bit was wrong? Was it in some way unreasonable for me to express my opinion that PRA is consistent? Have you in fact refuted it?
>
> Please read my statement carefully. I didn't say it's unreasonable for
> anyone to express an opinion!

I did read your statement carefully, and the statement was "you're simply wrong". So I asked you to tell me which bit was wrong, directly quoting what I had actually written. What I wrote was "It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it." Was that wrong, or not? If it wasn't wrong, then I guess you'd better retract your statement that I was simply wrong.

> What I said is "logically, one's own
> opinion wouldn't", and that's not the same.

Right, you said my opinion doesn't count. Well, I already discussed the issue of whether I could prove my claim. I said, yes, I can prove it from certain axioms, but that's neither here nor there because such a proof would carry no epistemological weight with you if you had genuine doubts about the matter. So I've already made it clear that if you have genuine doubts, I haven't got anything that would convince you. Do you in fact have genuine doubts? I'm sure this is at least the third time I've asked. I'm still patiently waiting for an answer. Are you unsure about whether PRA is consistent? Are you unsure about whether Q is consistent? Please answer. Thank you.

> >> What is expected is meta level proof of consistency [based on FOL(=)
> >> definition of consistency] which no one can produce: FOL(=) definition
> >> of consistency WILL NOT grant one a way to prove such consistency,
>

You can prove that PRA is consistent, from the right axiom set. PA for example. But I already said that such proofs would not be epistemologically convincing to a skeptic.

> > I don't know why you think this.
>
> Because I've already explained it:
>
> >> FOL(=) definition of consistency WILL NOT grant one a way to prove
> >> such consistency,
>

Why not?

> There's so much one can explain, you know.
>

I find your attempts to argue for your point of view unconvincing.

> >> _unlike the case of inconsistency_ .
>
> >> Why, Rupert, couldn't you understand/acknowledge such a simple fact?
>
> > I suppose it must be something to do with my cognitive limitations.
>
> Tell you what. Can you prove ZFC is consistent, _using only_ its
> axioms, FOL(=) rules of inference, and _NOT using_ the concept
> of recursion, truths about the natural numbers, language-structure
> theoretical truth?
>

I don't think I understand the question. You want to know can I prove the consistency of ZFC in ZFC? The answer is no. But I've got a vibe that's not really what you were asking.

> Yes or No please.
>

As I say, I'm thinking I probably don't understand the question.

[Nam Nguyen]

>>> If you gave a proof of �Con(PRA) from axioms which we generally recognised to be such that there was pretty compelling reason for believing them to be true, then that would be a proof that would carry some weight.

>>
>> But what is the formal definition of "pretty compelling" and "believe"
>> within FOL(=) framework? Isn't it true it's "pretty compelling" to
>> someone here in sci.logic to "believe" a number greater than 10^500
>> _is_ an infinite number?
>>
>
> The concepts don't have formal definitions. If someone believes that there is pretty compelling reason to believe that 10^500+1 is infinite, then they're wrong. We're not claiming that our beliefs about such matters are infallible.
>
>>>>> It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.
>>
>>>> I'm sorry Rupert that I don't know how else I could convey to you
>>>> that you're simply wrong here: logically, one's own opinion wouldn't
>>>> count.
>>
>>> You're saying I'm wrong. Which bit was wrong? Was it in some way unreasonable for me to express my opinion that PRA is consistent? Have you in fact refuted it?
>>
>> Please read my statement carefully. I didn't say it's unreasonable for
>> anyone to express an opinion!
>
> I did read your statement carefully, and the statement was "you're simply wrong". So I asked you to tell me which bit was wrong, directly quoting what I had actually written. What I wrote was "It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it." Was that wrong, or not? If it wasn't wrong, then I guess you'd better retract your statement that I was simply wrong.

You were wrong because you challenged me to refute your "belief"
PRA is consistent. It's logically wrong to assert PRA is consistent
based on "belief" and it's equally logically wrong to challenge some
one to refute a belief.

>
>> What I said is "logically, one's own
>> opinion wouldn't", and that's not the same.
>
> Right, you said my opinion doesn't count. Well, I already discussed the issue of whether I could prove my claim. I said, yes, I can prove it from certain axioms, but that's neither here nor there because such a proof would carry no epistemological weight with you if you had genuine doubts about the matter.

So from what axiom-set did you prove the consistency of PRA?

> So I've already made it clear that if you have genuine doubts, I haven't got anything that would convince you. Do you in fact have genuine doubts? I'm sure this is at least the third time I've asked. I'm still patiently waiting for an answer. Are you unsure about whether PRA is consistent? Are you unsure about whether Q is consistent? Please answer. Thank you.

It's not a question of my doubting anything. I have the right to doubt
certain things. The issue is if you prove the consistency of PRA in
some axiom-set called T, how would you prove _IN META LEVEL_ such T
is consistent, _USING THE VALID DEFINITION OF CONSISTENCY_to begin with?

(I've repeated this many many times already so I hope you don't mind
my being frank and say that I feel you're playing some kind of an
understanding-game with me.)

>
>>>> What is expected is meta level proof of consistency [based on FOL(=)
>>>> definition of consistency] which no one can produce: FOL(=) definition
>>>> of consistency WILL NOT grant one a way to prove such consistency,
>>
>
> You can prove that PRA is consistent, from the right axiom set. PA for example. But I already said that such proofs would not be epistemologically convincing to a skeptic.

What is the formal definition of "the right axiom set"? Apparently
it'd exclude the inconsistent axiom-sets, but what else would this
undefined phrase exclude, or include?

That's one of your problems here, Rupert: you haven't formally defined
what "the right axiom set" would logically mean, for proving the
consistency of a general T.

Isn't "the right axiom set" a private notion of yours and not a common
technical phrase defined within FOL(=)?

>
>>> I don't know why you think this.
>>
>> Because I've already explained it:
>>
>> >> FOL(=) definition of consistency WILL NOT grant one a way to prove
>> >> such consistency,
>>
>
> Why not?

That's why you should have responded to my requesting you to cite the
standard definition of consistency here. Would you cite it now, before
I re-explain the "Why not?" for you?

>
>> There's so much one can explain, you know.
>>
>
> I find your attempts to argue for your point of view unconvincing.
>
>>>> _unlike the case of inconsistency_ .
>>
>>>> Why, Rupert, couldn't you understand/acknowledge such a simple fact?
>>
>>> I suppose it must be something to do with my cognitive limitations.
>>
>> Tell you what. Can you prove ZFC is consistent, _using only_ its
>> axioms, FOL(=) rules of inference, and _NOT using_ the concept
>> of recursion, truths about the natural numbers, language-structure
>> theoretical truth?
>>
>
> I don't think I understand the question. You want to know can I prove the consistency of ZFC in ZFC? The answer is no. But I've got a vibe that's not really what you were asking.

Could you share with us why that's a no? (Please don't cite GIT,
per my stipulation above).

[George Greene]

On Sunday, November 24, 2013 1:43:47 AM UTC-5, Nam Nguyen wrote:
> Seriously, George, you're here not better at all compared to the crank
>
> you've on and off argued with: your reasoning and that of those crank
>
> are of the same pathetic style!

Nam, you're a crank. I'm not.
Your judgments are not relevant.

[George Greene]

On Sunday, November 24, 2013 1:38:49 AM UTC-5, Nam Nguyen wrote:
> >> Open question or not, one certainly can prove PRA is inconsistent.

I replied,

> > Unfortunately for you, however, that "one" IS NOT *you*.

NamN then had the sheer idiocy to reply,

> I'm a little bit sick and tired of Inquisition style of responding
> to my post, so why don't _you_ re-read my post and understand why

Dipshit: I DID read the part where you said "one certainly can prove
PRA is inconsistent". I am nowhere near "inquisition" style when
I flush that as the horse-shit that it is. OBVIOUSLY,NOBODY EVER HAS OR
EVER WILL prove that PRA is inconsistent. Even if you disagree with
the "never will" part, ONE CERTAINLY may NOT disagree with the "never has"
part! YOU HAVE NEVER SEEN a proof of the inconsistency of PRA!
GOOD GRIEF! The natural numbers ARE A MODEL of PRA, for starters!
You are just flaunting BASIC ignorance OF SIMPLE topics.
This is almost as bad as when you didn't know that natural numbers
are FINITE, BY DEFINITION (that is still sort of an all-time level
of ignorance from anyone who pretends NOT to be a crank).
Between JUST THOSE two sentences, you have reduced your credibility
in THIS forum BELOW ZERO. At a bare minimum, to get anything other
than abuse in response TO ANYthing, you will have to steer the discussion
AWAY from those two topics, because as soon as you get near them, it will
again become relevant that you thought PRA might be inconsistent
(without being anywhere near deriving a contradiction from it) or that
a natural number might be infinite (without being anywhere near -- which
is far worse -- A DICTIONARY).

[Nam Nguyen]

On 24/11/2013 6:02 PM, George Greene wrote:
> On Sunday, November 24, 2013 1:38:49 AM UTC-5, Nam Nguyen wrote:
>>>> Open question or not, one certainly can prove PRA is inconsistent.
>
>
> I replied,
>>> Unfortunately for you, however, that "one" IS NOT *you*.
>
>
> NamN then had the sheer idiocy to reply,
>> I'm a little bit sick and tired of Inquisition style of responding
>> to my post, so why don't _you_ re-read my post and understand why
>
> Dipshit: I DID read the part where you said "one certainly can prove
> PRA is inconsistent". I am nowhere near "inquisition" style when
> I flush that as the horse-shit that it is. OBVIOUSLY,NOBODY EVER HAS OR
> EVER WILL prove that PRA is inconsistent. Even if you disagree with
> the "never will" part, ONE CERTAINLY may NOT disagree with the "never has"
> part! YOU HAVE NEVER SEEN a proof of the inconsistency of PRA!

Not even _a proof_ ? What about (PRA + {~x=x}) |- ~CON(PRA)? Doesn't
that count as _a proof_ of inconsistency of PRA? If not, why?

> GOOD GRIEF! The natural numbers ARE A MODEL of PRA, for starters!

Good grief! George doesn't even know the truth value of cGC in the
natural numbers, so obviously George doesn't know what the natural
numbers are, for starters!

> You are just flaunting BASIC ignorance OF SIMPLE topics.

You, George, are just flaunting BASIC ignorance OF SIMPLE topics.

> This is almost as bad as when you didn't know that natural numbers
> are FINITE, BY DEFINITION (that is still sort of an all-time level
> of ignorance from anyone who pretends NOT to be a crank).

That makes you an Inquisition person: I did explain clearly and
correctly why natural numbers are genuinely neither finite nor infinite!

> Between JUST THOSE two sentences, you have reduced your credibility
> in THIS forum BELOW ZERO.

That's what Galileo got told too by the Inquisition, I'm sure.

What's news?

[Nam Nguyen]

On 24/11/2013 5:56 PM, George Greene wrote:
> On Sunday, November 24, 2013 1:43:47 AM UTC-5, Nam Nguyen wrote:
>> Seriously, George, you're here not better at all compared to the crank
>>
>> you've on and off argued with: your reasoning and that of those crank
>>
>> are of the same pathetic style!
>
> Nam, you're a crank. I'm not.

George, you're a modern relic of the Inquisition, of course.

> Your judgments are not relevant.

To the Inquisition, yes that's right!

[Nam Nguyen]

On 24/11/2013 12:00 PM, Nam Nguyen wrote:
> On 24/11/2013 11:26 AM, Rupert wrote:
>> On Sunday, November 24, 2013 7:02:53 PM UTC+1, Nam Nguyen wrote:
>>> On 24/11/2013 9:02 AM, Rupert wrote:
>>>> On Saturday, November 23, 2013 11:25:35 PM UTC+1, Nam Nguyen wrote:

>>>>> What is expected is meta level proof of consistency [based on FOL(=)
>>>>> definition of consistency] which no one can produce: FOL(=) definition
>>>>> of consistency WILL NOT grant one a way to prove such consistency,
>>>
>>
>> You can prove that PRA is consistent, from the right axiom set. PA for
>> example. But I already said that such proofs would not be
>> epistemologically convincing to a skeptic.
>
> What is the formal definition of "the right axiom set"? Apparently
> it'd exclude the inconsistent axiom-sets, but what else would this
> undefined phrase exclude, or include?
>
> That's one of your problems here, Rupert: you haven't formally defined
> what "the right axiom set" would logically mean, for proving the
> consistency of a general T.
>
> Isn't "the right axiom set" a private notion of yours and not a common
> technical phrase defined within FOL(=)?

In addition, you said (above) PA can prove the consistency of PRA.
And before you had said PRA would prove the consistency of Q.

But PA is an extension of Q. So _isn't all what you've said here is_
_just circular_ ?

[Nam Nguyen]

Just think about that: as a theory, ML is indeed identical to ML +
(~x=x}. So, what would make anyone be 100% sure that PRA wouldn't
be identical to PRA + {~x=x}?

Rupert suggested PA would provide such 100% certainty, but as I've
just pointed out in the other post, that's just being circular.

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 23/11/2013 8:50 AM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 22/11/2013 3:56 AM, Alan Smaill wrote:
>>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>>
>>>>> On 21/11/2013 4:35 AM, Alan Smaill wrote:
>>>>>> Nam Nguyen <namduc...@shaw.ca> writes:
>>>>>>
>>>> ...

>>>>>>> Here's the definition of T having M as its model:
>>>>>>>
>>>>>>> (M is a model of T) df= (M is a structure) and
>>>>>>> (T is consistent) and
>>>>>>> ((S is a statement-theorem in T)
>>>>>>> => (M |= S)).
>>>>>>
>>>>>> That's *not* what everyone else uses as definition of model.
>>>>>
>>>>> You don't have a proof for that.
>>>>
>>>> It's not in any text I know.
>>>>
>>>> Can you point me to anywhere this definition appears, apart
>>>> from your own claim?
>>>
>>> I don't have to show you anything here: I was only challenging to prove
>>> your own statement.
>>
>> OK, so you do not know of anyone else using your proposed definition.
>> I take it, therefore, that you do not think that Shoenfield
>> gives your definition.
>
> Word to word, No. But then there's such a thing as equivalence of
> _different_ definitions. Isosceles triangle can be defined in term
> of "side" or "angle", correct?

There's such a think as taking a definition that nobodey else but
you uses, as far as either of us knows, and then complaining about it.

> More about Shoenfield's definition of model of a T below.
>
>>
>>>>> The only thing it matters
>>>>> is whether or not it's a technically correct definition,
>>>>> and it is.
>>>>>
>>>>>> Look at Shoenfield, for once.
>>>>>
>>>>> _You_ should look it at least more twice: his definition
>>>>> contains a circularity. And in any rate from there you can
>>>>> conclude my there's nothing technically wrong with my definition above.
>>>>
>>>> I have it in front of me.
>>>> His definition has no condition that T is consistent.
>>>>
>>>> "By a model of a theory T, we mean a structure for L(T) in which all
>>>> the non-logical axioms of T are valid."
>>>>
>>>> The definition of valid is defined purely in terms of structures
>>>> (no mention of proof rules).
>>>
>>> Is that the only sentence he wrote in that _short paragraph_ ?
>>> Would you be able to excerpt that short paragraph here?
>>
>> "By a model of a theory T, we mean a structure for L(T) in which all
>> the non-logical axioms of T are valid. A formula is valid in T if
>> it is valid in every model of T; equivalently, if it is a logical
>> consequence of the nonlogical axioms of T."
>
> Before examining the usefulness, validity, of a _circular definition_
> here, do you concede that his definition of a model of a theory T is
> circular, as I've just pointed out to you above (my "his definition
> contains a circularity")?

No, I don't.
Please explain -- preferably considering my comment on your earlier
claim below.

>> So, no reference to any proof system, or to (syntactic) consistency.
>
> "So, no reference to any proof system"?

And so syntactic consistency is irrelevant to his *definition*.

> I was about to ask what then
> you'd think his "logical consequence of the nonlogical axioms of T"
> means, but how about your conceding about my circularity-request
> above first.

You might like to address my comment below, on why your next claim
immediately below is wrong.

>>>>>> In your terms, here's something nore like the normal definition:
>>>>>>
>>>>>> (M is a model of T) df= (M is a structure) and
>>>>>> ((S is a statement-theorem in T)
>>>>>> => (M |= S)).
>>>>>>
>>>>>>
>>>>>>> This definition presupposes T is consistent, because otherwise ~Ax[x=x]
>>>>>>> would be true in M, a violation of language structure definition.
>>>>
>>>> Not at all; from the definition, we can work out that ~Ax[x=x] has
>>>> no models. And we can deduce that therefore it is inconsistent.
>>>> (Take any formula Form in the relevant language.
>>>> Any such formula is a logical consequence of ~Ax[x=x], since
>>>> it is true in all models of ~Ax[x=x], vacuously (since the set
>>>> of models is empty). Completeness tells us that Form is provable.)
>>>>

--
Alan Smaill

[Nam Nguyen]

You're mistaken. I was not complaining. I was telling you that for any
common FOL(=) definition, one author's _rendition_ of the definition
does _NOT_ mean there aren't equivalent definitions.

For any formula F, there's ~~F; for any meta statement S, there's
neg(neg(S)). So there are such things as equivalent statements.

My "obligation" here is to show you his definition and mine are
equivalent, and we'd go from there. So, you just have to concede
that there are equivalent definitions, before I further my explanations.
Take it or leave it. Do you concede on this point?

[Nam Nguyen]

Iow, making arguments about foundational issues is more than just
taking an author's phrasing literally, word by word: it'd require
some common knowledge/understanding of the matter.

Fwiw, for whatever the reasons, authors could/would "gloss" when
occasions arise.

[Alan Smaill]

Yes, I agree on that.

On the other hand you *did* complain about circularity in a definition
that *you* provided, a definition different from the one in Shoenfield.

> For any formula F, there's ~~F; for any meta statement S, there's
> neg(neg(S)). So there are such things as equivalent statements.
>
> My "obligation" here is to show you his definition and mine are
> equivalent, and we'd go from there. So, you just have to concede
> that there are equivalent definitions, before I further my explanations.
> Take it or leave it. Do you concede on this point?

Carry on, equivalent definitions do exist.

--
Alan Smaill

[Rupert]

On Monday, November 25, 2013 5:29:50 AM UTC+1, Nam Nguyen wrote:
>
> > Remember the Quinne's ML theory: he obviously first _believed_ it being
> > consistent, but it turned out to be inconsistent, and equivalent to
> > ML + (~x=x}!
>
> Just think about that: as a theory, ML is indeed identical to ML +
> (~x=x}. So, what would make anyone be 100% sure that PRA wouldn't
> be identical to PRA + {~x=x}?
>
> Rupert suggested PA would provide such 100% certainty, but as I've
> just pointed out in the other post, that's just being circular.
>

No I didn't. There was never any claim to infallibility.

[Rupert]

> >>> If you gave a proof of �Con(PRA) from axioms which we generally recognised to be such that there was pretty compelling reason for believing them to be true, then that would be a proof that would carry some weight.

>
> >> But what is the formal definition of "pretty compelling" and "believe"
> >> within FOL(=) framework? Isn't it true it's "pretty compelling" to
> >> someone here in sci.logic to "believe" a number greater than 10^500
> >> _is_ an infinite number?
>
> > The concepts don't have formal definitions. If someone believes that there is pretty compelling reason to believe that 10^500+1 is infinite, then they're wrong. We're not claiming that our beliefs about such matters are infallible.
>
> >>>>> It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.
>
> >>>> I'm sorry Rupert that I don't know how else I could convey to you
> >>>> that you're simply wrong here: logically, one's own opinion wouldn't
> >>>> count.
>
> >>> You're saying I'm wrong. Which bit was wrong? Was it in some way unreasonable for me to express my opinion that PRA is consistent? Have you in fact refuted it?
>
> >> Please read my statement carefully. I didn't say it's unreasonable for
> >> anyone to express an opinion!
>
> > I did read your statement carefully, and the statement was "you're simply wrong". So I asked you to tell me which bit was wrong, directly quoting what I had actually written. What I wrote was "It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it." Was that wrong, or not? If it wasn't wrong, then I guess you'd better retract your statement that I was simply wrong.
>
> You were wrong because you challenged me to refute your "belief"
> PRA is consistent.

No, I didn't. You said you could prove that PRA is inconsistent, and I asked you to show me the proof.

> It's logically wrong to assert PRA is consistent
> based on "belief"

No, it's not.

> and it's equally logically wrong to challenge some
> one to refute a belief.
>

You made a claim and I asked you to support it. Nothing wrong with that.

> >> What I said is "logically, one's own
> >> opinion wouldn't", and that's not the same.
>
> > Right, you said my opinion doesn't count. Well, I already discussed the issue of whether I could prove my claim. I said, yes, I can prove it from certain axioms, but that's neither here nor there because such a proof would carry no epistemological weight with you if you had genuine doubts about the matter.
>
> So from what axiom-set did you prove the consistency of PRA?
>

I haven't given you a proof. You can prove it in PRA+"Ackermann's function is total", or PA. But that's neither here nor there. I already told you (at least twice now) that such proofs won't be epistemologically convincing to a skeptic.

> > So I've already made it clear that if you have genuine doubts, I haven't got anything that would convince you. Do you in fact have genuine doubts? I'm sure this is at least the third time I've asked. I'm still patiently waiting for an answer. Are you unsure about whether PRA is consistent? Are you unsure about whether Q is consistent? Please answer. Thank you.
>
> It's not a question of my doubting anything. I have the right to doubt
> certain things. The issue is if you prove the consistency of PRA in
> some axiom-set called T, how would you prove _IN META LEVEL_ such T
> is consistent, _USING THE VALID DEFINITION OF CONSISTENCY_to begin with?
>

I don't think I know what this means, "prove in meta-level".

I wish you would just answer the question.

> (I've repeated this many many times already so I hope you don't mind
> my being frank and say that I feel you're playing some kind of an
> understanding-game with me.)
>

I'm sorry you feel that way.

> >>>> What is expected is meta level proof of consistency [based on FOL(=)
> >>>> definition of consistency] which no one can produce: FOL(=) definition
> >>>> of consistency WILL NOT grant one a way to prove such consistency,
>
> > You can prove that PRA is consistent, from the right axiom set. PA for example. But I already said that such proofs would not be epistemologically convincing to a skeptic.
>
> What is the formal definition of "the right axiom set"?

It's not a formally defined concept.

> Apparently
> it'd exclude the inconsistent axiom-sets, but what else would this
> undefined phrase exclude, or include?
>
> That's one of your problems here, Rupert: you haven't formally defined
> what "the right axiom set" would logically mean, for proving the
> consistency of a general T.
>

And you haven't defined "prove in meta-level".

Why is it a problem?

> Isn't "the right axiom set" a private notion of yours and not a common
> technical phrase defined within FOL(=)?
>
> >>> I don't know why you think this.
>
> >> Because I've already explained it:
>
> >> >> FOL(=) definition of consistency WILL NOT grant one a way to prove
> >> >> such consistency,
>
> > Why not?
>
> That's why you should have responded to my requesting you to cite the
> standard definition of consistency here. Would you cite it now, before
> I re-explain the "Why not?" for you?
>

You just want me to give you the standard definition from memory, or do you want an actual page reference? I can't give you a page reference at the moment, I'm in the Netherlands.

> >> There's so much one can explain, you know.
>
> > I find your attempts to argue for your point of view unconvincing.
>
> >>>> _unlike the case of inconsistency_ .
>
> >>>> Why, Rupert, couldn't you understand/acknowledge such a simple fact?
>
> >>> I suppose it must be something to do with my cognitive limitations.
>
> >> Tell you what. Can you prove ZFC is consistent, _using only_ its
> >> axioms, FOL(=) rules of inference, and _NOT using_ the concept
> >> of recursion, truths about the natural numbers, language-structure
> >> theoretical truth?
>
> > I don't think I understand the question. You want to know can I prove the consistency of ZFC in ZFC? The answer is no. But I've got a vibe that's not really what you were asking.
>
> Could you share with us why that's a no? (Please don't cite GIT,
> per my stipulation above).
>

Why not? GIT is the reason.

[George Greene]

On Sunday, November 3, 2013 6:43:42 PM UTC-5, fom wrote:

> In spite of George's wealth of knowledge, he has
> certain non-standard views.

Unfortunately for you, the one you are about to cite IS NOT one of them.
And, obviously, *I* prefer to think that I hold non-standard
views BECAUSE of my knowledge, not in spite of it.

> For example, there is a notion (about which I have
> reservations) that the "definition"
> A bachelor is an unmarried man
> is an analytic statement.

No, I'm sorry; that IS NOT the notion.
It is A FACT and NOT a notion that "bachelor" IS DEFINED
as "unmarried man" (albeit among other definitions)
"in natural language dictionaries". A definition IS NOT
a "statement", analytic OR OTHERWISE, typically.
A definition is A DEFINITION.

> George will appeal to natural language dictionaries
> and refer to such definitions on the basis of their
> analyticity.

No, you are MIS-LOCATING the analyticty. The STATEMENT is what
gets to be analytic (or not). The definition (NOT being a statement)
DOESN'T. Analyticity is a form of necessity and one reason why definitions
cannot BE analytic is THAT ANY "thing"-that-gets-defined ARGUABLY COULD
have been defined DIFFERENTLY. "And" necessarily means what it means but
it didn't necessarily have to be CALLED "And". Just what two things ANY
definition is linking or associating is something that isn't normally
scrutinized BECAUSE SOME THINGS HAVE to be taken as just given WITHOUT
further justification/scrutiny, on pain of infinite regress.

Analyticity of the statement occurs AFTER the definitions, as a
CONSEQUENCE OF the definitions, because defined terms can be
legitimately interpreted to mean the things-they-are-defined-AS --
one can legitimately interpret any "definiendum" as "co-Sinnate"
(Fregean Sinn/sense) with ITS CORRESPONDING "definiens".
Mixing too mach German/Latin/English but YOU GET MY DRIFT.

[Peter Percival]

Rupert wrote:
> On Sunday, November 24, 2013 8:00:00 PM UTC+1, Nam Nguyen wrote:

>> That's why you should have responded to my requesting you to cite
>> the standard definition of consistency here. Would you cite it now,
>> before I re-explain the "Why not?" for you?
>>
>
> You just want me to give you the standard definition from memory, or
> do you want an actual page reference?

Shoenfield page 42.

> I can't give you a page
> reference at the moment, I'm in the Netherlands.

What is the formal definition of "I'm in the Netherlands"?

--
Madam Life's a piece in bloom,
Death goes dogging everywhere:
She's the tenant of the room,
He's the ruffian on the stair.

[James Burns]

On 11/26/2013 5:36 AM, Peter Percival wrote:
> Rupert wrote:
>> On Sunday, November 24, 2013 8:00:00 PM UTC+1,
>> Nam Nguyen wrote:

>>> That's why you should have responded to my requesting you to cite
>>> the standard definition of consistency here. Would you cite it now,
>>> before I re-explain the "Why not?" for you?
>>
>> You just want me to give you the standard definition from memory,
>> or do you want an actual page reference?
>
> Shoenfield page 42.
>
>> I can't give you a page
>> reference at the moment, I'm in the Netherlands.
>
> What is the formal definition of "I'm in the Netherlands"?

But first, what is the formal definition of "What is the formal
definition of 'I'm in the Netherlands'?"?

[Peter Olcott]

{Request} {FormalSemaniticSpecificationOf} {EnglishUtterance}:

"I'm in the Netherlands"

{IndividualPerson} {Within}(Predicate)
{GeoGraphicBoundariesOf}{NamedGeographicLocation}

[Nam Nguyen]

On 26/11/2013 3:36 AM, Peter Percival wrote:
> Rupert wrote:
>> On Sunday, November 24, 2013 8:00:00 PM UTC+1, Nam Nguyen wrote:
>
>>> That's why you should have responded to my requesting you to cite
>>> the standard definition of consistency here. Would you cite it now,
>>> before I re-explain the "Why not?" for you?
>>>
>>
>> You just want me to give you the standard definition from memory, or
>> do you want an actual page reference?
>
> Shoenfield page 42.

Right. So how many theories would you or Rupert see mentioned in each
of Shoenfield's definitions of inconsistency and consistency?

[Nam Nguyen]

Ok. So, is this the axiom:

x<y \/ x=y < y<x

(Shoenfield pg. 22) _true or false_ in the standard language structure
for the language of arithmetic?

[Remember according to you no rules of inference would be used in the
definition (your "no reference to any proof system,")].

[fom]

On 11/26/2013 4:36 AM, Peter Percival wrote:
> Rupert wrote:
>> On Sunday, November 24, 2013 8:00:00 PM UTC+1, Nam Nguyen wrote:
>
>>> That's why you should have responded to my requesting you to cite
>>> the standard definition of consistency here. Would you cite it now,
>>> before I re-explain the "Why not?" for you?
>>>
>>
>> You just want me to give you the standard definition from memory, or
>> do you want an actual page reference?
>
> Shoenfield page 42.
>
>> I can't give you a page
>> reference at the moment, I'm in the Netherlands.
>
> What is the formal definition of "I'm in the Netherlands"?
>

As it contains in indexical, it may require
Kaplan's logic of demonstratives.

:-)

http://plato.stanford.edu/entries/propositions-singular/#ReaForSinProArgIndDem

http://plato.stanford.edu/entries/indexicals/

[fom]

On 11/26/2013 7:11 AM, James Burns wrote:

"What does it mean to say logic is formal?"

http://johnmacfarlane.net/dissertation.pdf

:-)

http://plato.stanford.edu/entries/definitions/

:-)

I am getting an entire morning's worth of review
out of Peter's joke!

[Alan Smaill]

You mean I suppose:

x<y \/ x=y \/ y<x.

> (Shoenfield pg. 22) _true or false_ in the standard language structure
> for the language of arithmetic?
>
> [Remember according to you no rules of inference would be used in the
> definition (your "no reference to any proof system,")].

As Shoenfield says, example 1, p 23, the axiom holds in the standard model,

Now, where is the circularity?

--
Alan Smaill

[George Greene]

On Tuesday, November 26, 2013 5:36:05 AM UTC-5, Peter Percival wrote:
> What is the formal definition of "I'm in the Netherlands"?

The Netherlands CAN'T have a FORMAL definition BECAUSE it is a PHYSICAL place.
"I" doesn't NEED a "formal" definition -- it's an indexical -- it's defined in terms of the people having the conversation -- it refers to the entity uttering the utterance. "Am" is just metaphorical anyhow. If you were to translate this into formal language, "am" would very likely COMPLETELY DISAPPEAR.

[George Greene]

On Tuesday, November 26, 2013 11:46:48 AM UTC-5, fom wrote:
> As it contains in indexical, it may require
>
> Kaplan's logic of demonstratives.

Indexicals and demonstratives DON'T *REQUIRE* anything!
They are BASIC -- that's kind of the whole point! IF you were going
to explain or define ANYthing then the point at which YOU STOP would be
indexicals and demonstratives!

[George Greene]

On Tuesday, November 26, 2013 11:46:48 AM UTC-5, fom wrote:

> http://plato.stanford.edu/entries/indexicals/


The StEP article IS LAME. That Kaplan or anyone else would have chosen
to do this WITHOUT trying to get the sense/reference distinction (which long
precedes Kaplan's) right is deplorable. The article brings it up briefly but dsmisses it clumsily.

[Nam Nguyen]

Yes. (Thanks).

>
>
>> (Shoenfield pg. 22) _true or false_ in the standard language structure
>> for the language of arithmetic?
>>
>> [Remember according to you no rules of inference would be used in the
>> definition (your "no reference to any proof system,")].
>
> As Shoenfield says, example 1, p 23, the axiom holds in the standard model,
>
> Now, where is the circularity?

Hold on a second, Alan. We haven't been through yet with the axiom
(x<y \/ x=y \/ y<x) yet. _How would you know THIS axiom is true_
in the standard structure?

I know you mentioned pg 23 but that page does mention only general
axioms, not a specific one like (x<y \/ x=y \/ y<x).

(Read: I've given you the caveat that there might be "glossing" in
some author's books but you seem to have ignored. You're on your own!)

[Nam Nguyen]

>>>>> If you gave a proof of �Con(PRA) from axioms which we generally recognised to be such that there was pretty compelling reason for believing them to be true, then that would be a proof that would carry some weight.

>>
>>>> But what is the formal definition of "pretty compelling" and "believe"
>>>> within FOL(=) framework? Isn't it true it's "pretty compelling" to
>>>> someone here in sci.logic to "believe" a number greater than 10^500
>>>> _is_ an infinite number?
>>
>>> The concepts don't have formal definitions. If someone believes that there is pretty compelling reason to believe that 10^500+1 is infinite, then they're wrong. We're not claiming that our beliefs about such matters are infallible.
>>
>>>>>>> It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it.
>>
>>>>>> I'm sorry Rupert that I don't know how else I could convey to you
>>>>>> that you're simply wrong here: logically, one's own opinion wouldn't
>>>>>> count.
>>
>>>>> You're saying I'm wrong. Which bit was wrong? Was it in some way unreasonable for me to express my opinion that PRA is consistent? Have you in fact refuted it?
>>
>>>> Please read my statement carefully. I didn't say it's unreasonable for
>>>> anyone to express an opinion!
>>
>>> I did read your statement carefully, and the statement was "you're simply wrong". So I asked you to tell me which bit was wrong, directly quoting what I had actually written. What I wrote was "It was perfectly reasonable of me to express my opinion that PRA is consistent, and obviously you haven't refuted it." Was that wrong, or not? If it wasn't wrong, then I guess you'd better retract your statement that I was simply wrong.
>>
>> You were wrong because you challenged me to refute your "belief"
>> PRA is consistent.
>
> No, I didn't. You said you could prove that PRA is inconsistent, and I asked you to show me the proof.
>
>> It's logically wrong to assert PRA is consistent
>> based on "belief"
>
> No, it's not.

Well then you aren't talking about mathematical logic, and that's why
you wouldn't be able to understand my proof in this thread (and any
related ones).

We could exchange a million posts but until you admit you're wrong
in saying that you could _logically assert_ the consistency of a formal
system using belief, it wouldn't change the impasse caused by you.

[Nam Nguyen]

To be succinct, how would you know the _axiom_ , say, (m < n) is true?

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> To be succinct, how would you know the _axiom_ , say, (m < n) is true?

You wouldn't, because it's not true --
Shoenfield's definition treats variables as implicitly universally quantified.

--
Alan Smaill

[Rupert]

On Wednesday, November 27, 2013 3:01:12 AM UTC+1, Nam Nguyen wrote:
> Well then you aren't talking about mathematical logic, and that's why
> you wouldn't be able to understand my proof in this thread (and any
> related ones).
>
> We could exchange a million posts but until you admit you're wrong
> in saying that you could _logically assert_ the consistency of a formal
> system using belief, it wouldn't change the impasse caused by you.
>

I suppose, if you choose, you can try to offer me reasons why I should come to the conclusion that I was wrong.

[Jim Burns]

On 11/26/2013 9:01 PM, Nam Nguyen wrote:
> On 25/11/2013 3:30 PM, Rupert wrote:
>> On Sunday, November 24, 2013 8:00:00 PM UTC+1, Nam Nguyen wrote:

[...]

>>> You were wrong because you challenged me to refute your "belief"
>>> PRA is consistent.
>>
>> No, I didn't. You said you could prove that PRA is inconsistent, and I
>> asked you to show me the proof.
>>
>>> It's logically wrong to assert PRA is consistent
>>> based on "belief"
>>
>> No, it's not.
>
> Well then you aren't talking about mathematical logic, and that's why
> you wouldn't be able to understand my proof in this thread (and any
> related ones).
>
> We could exchange a million posts but until you admit you're wrong
> in saying that you could _logically assert_ the consistency of a formal
> system using belief, it wouldn't change the impasse caused by you.

Rupert said that he could _logically assert_ the consistency of PRA
from various sets of axioms, among them PA. Note that the word
"believe" was not used when he said that.

(Paraphrasing) he also said that anyone skeptical of PRA would very
likely be skeptical as well of any axiom set from which PRA could
be proven. Let us refer to such a person as a radical skeptic.

The question to be asked of a radical skeptic is: If not PRA,
if not PA, what _will_ you accept, in order to prove theorems
about the natural numbers?

I remind you that you have been claiming to have theorems about
the natural numbers. It certainly _looks as though_ there should be
some set of axioms that you will accept in order to prove your
own theorems, but appearances can be deceiving.

What logical ground do you stand upon when you say that you
can prove various statements about cGC? Can anyone other than
you stand on that ground?

[Alan Smaill]

Nam Nguyen <namduc...@shaw.ca> writes:

> On 26/11/2013 10:31 AM, Alan Smaill wrote:
>> Nam Nguyen <namduc...@shaw.ca> writes:
>>
>>> On 25/11/2013 12:05 PM, Alan Smaill wrote:

...

>>>> Carry on, equivalent definitions do exist.
>>>
>>> Ok. So, is this the axiom:
>>>
>>> x<y \/ x=y < y<x
>>
>> You mean I suppose:
>>
>> x<y \/ x=y \/ y<x.
>
> Yes. (Thanks).
>
>>> (Shoenfield pg. 22) _true or false_ in the standard language structure
>>> for the language of arithmetic?
>>>
>>> [Remember according to you no rules of inference would be used in the
>>> definition (your "no reference to any proof system,")].
>>
>> As Shoenfield says, example 1, p 23, the axiom holds in the standard model,
>>
>> Now, where is the circularity?
>
> Hold on a second, Alan. We haven't been through yet with the axiom
> (x<y \/ x=y \/ y<x) yet. _How would you know THIS axiom is true_
> in the standard structure?
>
> I know you mentioned pg 23 but that page does mention only general
> axioms, not a specific one like (x<y \/ x=y \/ y<x).

Since you presumably have the book, you see that example 1 gives
a model for |N, which in turn is defined on p 22 to include
the specific axiom in question.

You can follow through Shoenfield's definition of being a model
and see if it holds in that structure, by meta-theoretic reasoning.

Now, where is the circularity?

--
Alan Smaill

[Nam Nguyen]

I see: he implicitly alluded to it in example 1. I was referring to
a portion of his proof for the Validity Theorem which he mentioned
a general nonlogical axiom. Ayy rate ...

>
> You can follow through Shoenfield's definition of being a model
> and see if it holds in that structure, by meta-theoretic reasoning.
>
> Now, where is the circularity?

Not so fast, as I've mentioned, you still have _NOT_ yet explained
why the axiom (x<y \/ x=y \/ y<x) is true in the standard language
structure, without mentioning rules of inference.

Please do explain so before I could explain the circularity.

Yes, you did say:

> Shoenfield's definition treats variables as implicitly universally
> quantified.

but he's dead so we can't tell whether in his mind he might have
"treated" variable as such, or he might have meant the Generalization
Rule be used.

Technically of course the Generalization must be used, but it's you
who has to technically explain why this axiom is true.

[Nam Nguyen]

Sure. Although I think I did in one way or the other from time to time.

Basically, to assert a formal system T (PRA in this example) is
consistent, on the basis of individual belief (your "I believe
that it's consistent" in this example) is to disregard the technical
definitions of consistency, and hence the assertion is invalid: i.e.
wrong.

Here's our conversation before:

Nam asked Rupert:

>> Or are you just assuming PRA is consistent?

Rupert responded:

> I believe that it's consistent. If you're asking "Can you prove it",
> then that would obviously depend on which axioms you're going to
> allow me to use. But that's neither here nor there, really, because
> if you've got genuine doubts about the consistency of PRA then any
> proof that I can give you is not going to be epistemologically
> convincing for you.

Basically to you it seems the assertion of the alleged consistency
of PRA is _a sheer matter of either "belief" or "doubt"_ and _the_
_definition of consistency is disregarded_ .

Again, to what I gather, that's _not_ mathematical logic and is
simply wrong.

[Note that it's my position all along that CON(T) would express the
assertion T being consistent, but _in no logical manner would be_
_equated to the truth of the assertion itself_ .]

[Nam Nguyen]

Iow, in this debate you'd be wrong until you acknowledge that
consistency can't be known by definition of consistency, notwithstanding
that you could prove CON(T) in _another_ formal systems T'.

Iow, T' |- CON(T) can never be equated to T being consistent, logically
speaking.

[Nam Nguyen]

On 27/11/2013 5:00 AM, Jim Burns wrote:
> On 11/26/2013 9:01 PM, Nam Nguyen wrote:
>> On 25/11/2013 3:30 PM, Rupert wrote:
>>> On Sunday, November 24, 2013 8:00:00 PM UTC+1, Nam Nguyen wrote:
>
> [...]
>
>>>> You were wrong because you challenged me to refute your "belief"
>>>> PRA is consistent.
>>>
>>> No, I didn't. You said you could prove that PRA is inconsistent, and I
>>> asked you to show me the proof.
>>>
>>>> It's logically wrong to assert PRA is consistent
>>>> based on "belief"
>>>
>>> No, it's not.
>>
>> Well then you aren't talking about mathematical logic, and that's why
>> you wouldn't be able to understand my proof in this thread (and any
>> related ones).
>>
>> We could exchange a million posts but until you admit you're wrong
>> in saying that you could _logically assert_ the consistency of a formal
>> system using belief, it wouldn't change the impasse caused by you.
>
> Rupert said that he could _logically assert_ the consistency of PRA
> from various sets of axioms, among them PA. Note that the word
> "believe" was not used when he said that.

The problem is if he stays with strict definition of consistency
(which is sheer syntactical definition) he'd run into infinite
regression: how would he prove PA is consistent?

_Specifically which CONSISTENT formal system would he in turn_
_use to prove the consistency of PA_ ?

>
> (Paraphrasing) he also said that anyone skeptical of PRA would very
> likely be skeptical as well of any axiom set from which PRA could
> be proven. Let us refer to such a person as a radical skeptic.
>
> The question to be asked of a radical skeptic is: If not PRA,
> if not PA, what _will_ you accept, in order to prove theorems
> about the natural numbers?
>
> I remind you that you have been claiming to have theorems about
> the natural numbers. It certainly _looks as though_ there should be
> some set of axioms that you will accept in order to prove your
> own theorems, but appearances can be deceiving.
>
> What logical ground do you stand upon when you say that you
> can prove various statements about cGC? Can anyone other than
> you stand on that ground?

Sure. If it being impossible to assert the truth value of both
cGC and ~cGC is what we're talking about.

The logical ground is quite simple: given strictly less knowledge
that what an assertion would require, you can't logically make the
determination if the assertion is true or false.

For instance, if given only the information about the set S that
{} is a subset of S, it's impossible to assert S is empty, or
to assert S is not empty!

[Marshall]

On Wednesday, November 27, 2013 8:47:42 PM UTC-8, Nam Nguyen wrote:
>
> > Rupert said that he could _logically assert_ the consistency of PRA
> > from various sets of axioms, among them PA. Note that the word
> > "believe" was not used when he said that.
>
> The problem is if he stays with strict definition of consistency
> (which is sheer syntactical definition) he'd run into infinite
> regression: how would he prove PA is consistent?

Are you aware of any theories which you consider consistent?
Or is it your contention that there are no theories which are
known to be consistent?

My view: it is possible to know of some consistent theories.
For example, every equational theory* is consistent.


Marshall

* An equational theory is a theory whose axioms consist only
of equations (with variables implicitly universally quantified.)

[Nam Nguyen]

On 27/11/2013 10:30 PM, Marshall wrote:
> On Wednesday, November 27, 2013 8:47:42 PM UTC-8, Nam Nguyen wrote:
>>
>>> Rupert said that he could _logically assert_ the consistency of PRA
>>> from various sets of axioms, among them PA. Note that the word
>>> "believe" was not used when he said that.
>>
>> The problem is if he stays with strict definition of consistency
>> (which is sheer syntactical definition) he'd run into infinite
>> regression: how would he prove PA is consistent?
>
> Are you aware of any theories which you consider consistent?

But what exactly would you mean bey "aware": proof by definition of
consistency, or intuition?

> Or is it your contention that there are no theories which are
> known to be consistent?

It's my (meta) assertion that it's impossible to use FOL proof
machinery to prove actual consistency.

>
> My view: it is possible to know of some consistent theories.
> For example, every equational theory* is consistent.

That's an incorrect view. (Even if you have a finite language
structure in which you intuit the axioms be true, you still can
assert the consistency using FOL proof machinery, notwithstanding
that the underlying T might still be consistent.

The issue isn't whether or not T is consistent, but how logically
you could make meta level inference that it's consistent, from the
definition of consistency.

You simply can't: unlike the case of inconsistency.

You could change the definition of consistency, but that would
be another matter of course.

>
>
> Marshall
>
> * An equational theory is a theory whose axioms consist only
> of equations (with variables implicitly universally quantified.)

It doesn't matter: it's still an axiom-set in the final analysis,
whose (in)consistency is till governed by the definition of
(in)consistency, with all the advantage and baggage associated
with the definition.

[Rupert]

I don't have any idea why you would think the definition is disregarded.

> Again, to what I gather, that's _not_ mathematical logic and is
> simply wrong.
>

What do you *mean* when you say it's wrong? Do you mean that PRA is in fact inconsistent?

> [Note that it's my position all along that CON(T) would express the
> assertion T being consistent, but _in no logical manner would be_
> _equated to the truth of the assertion itself_ .]
>

I don't have any idea what that is supposed to mean.

[Rupert]

On Thursday, November 28, 2013 5:30:41 AM UTC+1, Nam Nguyen wrote:
>
> Iow, in this debate you'd be wrong until you acknowledge that
> consistency can't be known by definition of consistency, notwithstanding
> that you could prove CON(T) in _another_ formal systems T'.
>
> Iow, T' |- CON(T) can never be equated to T being consistent, logically
> speaking.
>

I don't have any idea why you think consistency can't be known.

[Nam Nguyen]

On 27/11/2013 10:55 PM, Nam Nguyen wrote:
> On 27/11/2013 10:30 PM, Marshall wrote:
>> On Wednesday, November 27, 2013 8:47:42 PM UTC-8, Nam Nguyen wrote:
>>>
>>>> Rupert said that he could _logically assert_ the consistency of PRA
>>>> from various sets of axioms, among them PA. Note that the word
>>>> "believe" was not used when he said that.
>>>
>>> The problem is if he stays with strict definition of consistency
>>> (which is sheer syntactical definition) he'd run into infinite
>>> regression: how would he prove PA is consistent?
>>
>> Are you aware of any theories which you consider consistent?
>
> But what exactly would you mean bey "aware": proof by definition of
> consistency, or intuition?
>
>> Or is it your contention that there are no theories which are
>> known to be consistent?
>
> It's my (meta) assertion that it's impossible to use FOL proof
> machinery to prove actual consistency.
>
>>
>> My view: it is possible to know of some consistent theories.
>> For example, every equational theory* is consistent.
>
> That's an incorrect view. (Even if you have a finite language

> structure in which you intuit the axioms be true, you still can't

> assert the consistency using FOL proof machinery, notwithstanding

> that the underlying T might still be consistent).

[I've made some typo correction in the above].

[Nam Nguyen]

We've gone back and forth a trillion times already, so I don't
mind going over it again.

But first, please do tell in full typing:

(A) what the definition of consistency of a T is.

And secondly, please do tell:

(B) what you'd mean by _knowing a T is consistent_ .
Iow, what criteria would you use to _know_
a general T be consistent.

[Nam Nguyen]

On 27/11/2013 10:57 PM, Nam Nguyen wrote:
> On 27/11/2013 10:55 PM, Nam Nguyen wrote:
>> On 27/11/2013 10:30 PM, Marshall wrote:
>>> On Wednesday, November 27, 2013 8:47:42 PM UTC-8, Nam Nguyen wrote:
>>>>
>>>>> Rupert said that he could _logically assert_ the consistency of PRA
>>>>> from various sets of axioms, among them PA. Note that the word
>>>>> "believe" was not used when he said that.
>>>>
>>>> The problem is if he stays with strict definition of consistency
>>>> (which is sheer syntactical definition) he'd run into infinite
>>>> regression: how would he prove PA is consistent?
>>>
>>> Are you aware of any theories which you consider consistent?
>>
>> But what exactly would you mean bey "aware": proof by definition of
>> consistency, or intuition?
>>
>>> Or is it your contention that there are no theories which are
>>> known to be consistent?
>>
>> It's my (meta) assertion that it's impossible to use FOL proof
>> machinery to prove actual consistency.

So, to answer your question, yes: it's true there are no formal
theories which are known to be consistent.

There are only formal theories which are intuited to be consistent.