Is Gödel's Incompleteness Theorem an invalid meta mathematical reasoning?

[Khong Dong]

Fwiw, I've posted (and answered) the question in MSE:

https://math.stackexchange.com/questions/2801432/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning

We'll see how that goes I'd guess.

[Rupert]

Just one small point you may be interested to know, your definition of \omega-consistency is not correct. What you have defined is closure under the \omega-rule. \Omega-consistency is when you do not have P(0), P(S0), P(SS0), ... and ~(An)P(n) all being theorems of a theory T.

[Rupert]

On Wednesday, May 30, 2018 at 7:42:34 AM UTC+2, Khong Dong wrote:

So yeah, hang on a second. You correctly observe that a rule inferring "There are infinitely many n such that P(n)" from P(0), P(S0), P(SS0), ... would be a valid rule, but a rule inferring "(An)P(n)" from "There are infinitely many n such that P(n)" would not be a valid rule. So fair enough.

Now, somehow from that you get that a rule inferring (An)P(n) from P(0), P(S0), P(SS0), ... is not a valid rule.

How, pray tell, does that follow?

[Khong Dong]

Under the context there's no difference in essence: both the hypothesis
hype(ω_consistency) and the conclusion conclude(ω_consistency) be true.

[Rupert]

There's a difference.

[Khong Dong]

Of course the set of infinite terms has to be less than or equal to the set
of infinite individuals which is less than or equal to the infinite set of
totality of individuals.

But "less than or equal to" doesn't exclude "strictly less than", right? And
in which case it's invalid to logically claim the first is the same as the
last. You couldn't see it?

[Khong Dong]

Not to my argument and not to the essence.

[Rupert]

Yeah, trying to converse with you isn't really all that profitable, is it?

You gave a definition of the notion of \omega-consistency which was flat-out wrong. I did my best to explain it to you.

[Khong Dong]

Think of what upper LST says.

[Rupert]

Yeah, your point being it's not a valid rule in non-standard models, sure. But that's got nothing to do with the reasoning you gave.

But yes, you are right. It is not a valid rule in non-standard models. However, this has no bearing whatsoever on Goedel's incompleteness theorem, because you got the definition of \omega-consistency wrong. Goedel has no need to assume the validity of the \omega-rule for arbitrary models of Peano Arithmetic.

[Khong Dong]

No it's just you're unable to understand a technical writing. That's all.

Stipulating H and C be true, is _in essence_ no difference than stipulating
(H => C) and H be true. In meta level.

[Rupert]

You defined closure under the \omega-rule, not \omega-consistency. They are different. It is really quite straightforward. The problem does not lie with any deficiency in my understanding.

[Khong Dong]

No. You're the one who isb't able to read the technical post there.

> Goedel has no need to assume the validity of the \omega-rule for arbitrary
> models of Peano Arithmetic.

Wrong. As presented by him, Goedel's paper requires \omega-consistency for
his GIT1 "proof".

[graham...@gmail.com]

Its not necessary.


In EXISTENTIAL SET THEORY there are no "FORALL(p)"



Here we prove 2 sets equivalent _WITHOUT_ using

EXTENTIONALITY x=y <-> Am mex<->mey






Consider smalls = { X | EXISTS(B) B>X }

then prove EXISTS(C) C in NATS ^ !C in SMALLS

yet if C in NATS then C+1 in NATS

so EXISTS(B) B=C+1 B>C

therefore C in smalls

[Rupert]

I see. So the problem lies entirely with me. Well, you know, good luck with finding people who are on the same level as you, so to speak.

[graham...@gmail.com]

try something simpler then Rudert

g = sqrt(3)*10^35

[Khong Dong]

On Wednesday, 30 May 2018 01:33:24 UTC-6, Khong Dong wrote:

Now on your part, how would you go from all the terms 0, S0, SS0, ... satisfying P
to _all_ individuals satisfying P? *Which logic rules or laws of thought did you use*?
Or you actually didn't use any logic rules or laws of thought at all?

[Rupert]

Hmmmm. You seem to want to continue the conversation.

It's a valid rule of inference for the standard model, but not for arbitrary non-standard models. As I believe I already mentioned.

Oh, and by the way, you got the definition of "\omega-consistency" wrong. You really should pay some heed to that part.

[Khong Dong]

On Wednesday, 30 May 2018 19:07:55 UTC-6, Rupert wrote:
> On Wednesday, May 30, 2018 at 6:45:49 PM UTC+2, Khong Dong wrote:
> > On Wednesday, 30 May 2018 01:33:24 UTC-6, Khong Dong wrote:
> > > On Wednesday, 30 May 2018 00:55:14 UTC-6, Rupert wrote:
> > > > On Wednesday, May 30, 2018 at 7:42:34 AM UTC+2, Khong Dong wrote:
> > > > > Fwiw, I've posted (and answered) the question in MSE:
> > > > >
> > > > > https://math.stackexchange.com/questions/2801432/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
> > > > >
> > > > > We'll see how that goes I'd guess.
> > > >
> > > > So yeah, hang on a second. You correctly observe that a rule inferring "There are infinitely many n such that P(n)" from P(0), P(S0), P(SS0), ... would be a valid rule, but a rule inferring "(An)P(n)" from "There are infinitely many n such that P(n)" would not be a valid rule. So fair enough.
> > > >
> > > > Now, somehow from that you get that a rule inferring (An)P(n) from P(0), P(S0), P(SS0), ... is not a valid rule.
> > > >
> > > > How, pray tell, does that follow?
> > >
> > > Of course the set of infinite terms has to be less than or equal to the set
> > > of infinite individuals which is less than or equal to the infinite set of
> > > totality of individuals.
> > >
> > > But "less than or equal to" doesn't exclude "strictly less than", right? And
> > > in which case it's invalid to logically claim the first is the same as the
> > > last. You couldn't see it?
> >
> > Now on your part, how would you go from all the terms 0, S0, SS0, ... satisfying P
> > to _all_ individuals satisfying P? *Which logic rules or laws of thought did you use*?
> > Or you actually didn't use any logic rules or laws of thought at all?
>
> Hmmmm. You seem to want to continue the conversation.

Oh no, you're mistaken. My question is a way to point out to sci.logic your
lack of understanding Godel's assumptions in his GIT1, whether or not you
answer it it doesn't matter: your lack of understanding so far is unchanged.

> It's a valid rule of inference for the standard model, but not for arbitrary
> non-standard models. As I believe I already mentioned.

Non-sequitur. As emphasized in my MSE post it was about what Godel wrote in
1931 and his omega-consistency was about the natural numbers not your buzzwords
"the standard model" or "non-standard model".

Again, his omega-consistency stipulation is invalid as pointed out in details
in the MSE post. Go and read it again: hiding beside the buzzwords "the
standard model" and "non-standard model" won't help you understand the issue.

> Oh, and by the way, you got the definition of "\omega-consistency" wrong. You really should pay some heed to that part.

It's only your delusion. (I didn't get it wrong, you're just not competent
enough to parse my MSE analysis on Godel's stipulations about omega
consistency in his GIT1.)

[Khong Dong]

My question, again: *Which logic rules or laws of thought did you use*?

And so far you're unable to answer it!

_Which_ logic rules or laws of thought did you use, Rupert?

[Rupert]

On Thursday, May 31, 2018 at 4:13:01 AM UTC+2, Khong Dong wrote:
> On Wednesday, 30 May 2018 19:07:55 UTC-6, Rupert wrote:
> > On Wednesday, May 30, 2018 at 6:45:49 PM UTC+2, Khong Dong wrote:
> > > On Wednesday, 30 May 2018 01:33:24 UTC-6, Khong Dong wrote:
> > > > On Wednesday, 30 May 2018 00:55:14 UTC-6, Rupert wrote:
> > > > > On Wednesday, May 30, 2018 at 7:42:34 AM UTC+2, Khong Dong wrote:
> > > > > > Fwiw, I've posted (and answered) the question in MSE:
> > > > > >
> > > > > > https://math.stackexchange.com/questions/2801432/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
> > > > > >
> > > > > > We'll see how that goes I'd guess.
> > > > >
> > > > > So yeah, hang on a second. You correctly observe that a rule inferring "There are infinitely many n such that P(n)" from P(0), P(S0), P(SS0), ... would be a valid rule, but a rule inferring "(An)P(n)" from "There are infinitely many n such that P(n)" would not be a valid rule. So fair enough.
> > > > >
> > > > > Now, somehow from that you get that a rule inferring (An)P(n) from P(0), P(S0), P(SS0), ... is not a valid rule.
> > > > >
> > > > > How, pray tell, does that follow?
> > > >
> > > > Of course the set of infinite terms has to be less than or equal to the set
> > > > of infinite individuals which is less than or equal to the infinite set of
> > > > totality of individuals.
> > > >
> > > > But "less than or equal to" doesn't exclude "strictly less than", right? And
> > > > in which case it's invalid to logically claim the first is the same as the
> > > > last. You couldn't see it?
> > >
> > > Now on your part, how would you go from all the terms 0, S0, SS0, ... satisfying P
> > > to _all_ individuals satisfying P? *Which logic rules or laws of thought did you use*?
> > > Or you actually didn't use any logic rules or laws of thought at all?
> >
> > Hmmmm. You seem to want to continue the conversation.
>
> Oh no, you're mistaken.

Oh good. See you then.

[Khong Dong]

Sure. Anyway you're still unable to answer the above question on how you could

go from all the terms 0, S0, SS0, ... satisfying P to _all_ individuals

satisfying P.

Not surprised at all.

[Rupert]

Well, hang on, do you want to continue the conversation or don't you... ???

The answer to your question is that that inference would be valid if the referents of the terms 0, S0, SS0, ... were all of the individuals in the domain of discourse, as is the case with the standard model. But, as I agreed, it would not be a valid inference in non-standard models.

[Khong Dong]

Sorry I'm not playing that game. I ask or say what I want and you respond or
not to your heart content. End of story.

>
> The answer to your question is that that inference would be valid if the referents of the terms 0, S0, SS0, ... were all of the individuals in the domain of discourse, as is the case with the standard model. But, as I agreed, it would not be a valid inference in non-standard models.

Well, see. You're not able to answer my question which among other things
doesn't mention the words "standard model" or "non-standard models".

Can you answer my questions without using those words?

[Rupert]

No, because you inaccurately paraphrased my position and I need to use "standard model" and "non-standard model" to explain what my actual position is.

[Rupert]

Okay, so here's your answer: Given that M is a model of Robinson arithmetic, it is not valid to infer from M satisfies P(0), P(S0), P(SS0), ... to M satisfies (An)P(n).

This is actually equivalent to saying that there exist theories that are consistent but not \omega-consistent, which is actually an observation that was first made by Goedel in his 1931 paper.

You're welcome.

[Julio Di Egidio]

On Thursday, 31 May 2018 10:21:21 UTC+2, Rupert wrote:
<snip>

> Okay, so here's your answer: Given that M is a model of Robinson arithmetic, it is not valid to infer from M satisfies P(0), P(S0), P(SS0), ... to M satisfies (An)P(n).

Yes, because in Robinson arithmetic everything is just finite, so that one
cannot even state that for-all.

> This is actually equivalent to saying that there exist theories that are consistent but not \omega-consistent

Is it? AFAIK, a theory is w-inconsistent if it *proves* exists n (nonstandard!)
such that P(n) *fails*. (E.g. as in the balls and vase problem...)

Julio

[Rupert]

On Thursday, May 31, 2018 at 11:13:55 AM UTC+2, Julio Di Egidio wrote:
> On Thursday, 31 May 2018 10:21:21 UTC+2, Rupert wrote:
> <snip>
> > Okay, so here's your answer: Given that M is a model of Robinson arithmetic, it is not valid to infer from M satisfies P(0), P(S0), P(SS0), ... to M satisfies (An)P(n).
>
> Yes, because in Robinson arithmetic everything is just finite, so that one
> cannot even state that for-all.

You couldn't have a formula of finite length stating the infinite conjunction, no, I was thinking of an infinitary rule of inference.

> > This is actually equivalent to saying that there exist theories that are consistent but not \omega-consistent
>
> Is it? AFAIK, a theory is w-inconsistent if it *proves* exists n (nonstandard!)
> such that P(n) *fails*. (E.g. as in the balls and vase problem...)

A theory T in the first-order language of arithmetic is said to be \omega-inconsistent if there is some formula P(x) with exactly one free variable, such that P(0), P(S0), P(SS0), ... ~(Ax)P(x), are all theorems of T.

[Julio Di Egidio]

On Thursday, 31 May 2018 11:20:07 UTC+2, Rupert wrote:
> On Thursday, May 31, 2018 at 11:13:55 AM UTC+2, Julio Di Egidio wrote:
> > On Thursday, 31 May 2018 10:21:21 UTC+2, Rupert wrote:
> > <snip>
> > > Okay, so here's your answer: Given that M is a model of Robinson arithmetic, it is not valid to infer from M satisfies P(0), P(S0), P(SS0), ... to M satisfies (An)P(n).
> >
> > Yes, because in Robinson arithmetic everything is just finite, so that one
> > cannot even state that for-all.
>
> You couldn't have a formula of finite length stating the infinite conjunction, no, I was thinking of an infinitary rule of inference.

Robison arithmetic is PA minus Infinity, as simple as that: "infinitary rule
of inference" is not even very meaningful.

> > > This is actually equivalent to saying that there exist theories that are consistent but not \omega-consistent
> >
> > Is it? AFAIK, a theory is w-inconsistent if it *proves* exists n (nonstandard!)
> > such that P(n) *fails*. (E.g. as in the balls and vase problem...)
>
> A theory T in the first-order language of arithmetic is said to be \omega-inconsistent if there is some formula P(x) with exactly one free variable, such that P(0), P(S0), P(SS0), ... ~(Ax)P(x), are all theorems of T.

Yes, which is NOT equivalent to not being able to infer etc. etc.

Never mind, you just had me puzzled for a moment.

Julio

[Rupert]

On Thursday, May 31, 2018 at 11:43:29 AM UTC+2, Julio Di Egidio wrote:
> On Thursday, 31 May 2018 11:20:07 UTC+2, Rupert wrote:
> > On Thursday, May 31, 2018 at 11:13:55 AM UTC+2, Julio Di Egidio wrote:
> > > On Thursday, 31 May 2018 10:21:21 UTC+2, Rupert wrote:
> > > <snip>
> > > > Okay, so here's your answer: Given that M is a model of Robinson arithmetic, it is not valid to infer from M satisfies P(0), P(S0), P(SS0), ... to M satisfies (An)P(n).
> > >
> > > Yes, because in Robinson arithmetic everything is just finite, so that one
> > > cannot even state that for-all.
> >
> > You couldn't have a formula of finite length stating the infinite conjunction, no, I was thinking of an infinitary rule of inference.
>
> Robison arithmetic is PA minus Infinity, as simple as that: "infinitary rule
> of inference" is not even very meaningful.

Finitary rule of inference is set of ordered pairs consisting of a finite set of formulas and a formula. If a pair is in the set then you can infer from the finite collection of formulas which is the first member of the pair, to the formula which is the second member of the pair. Infinitary rule of inference is the same, except that the set of formulas which is the first member of the pair is infinite.

[Julio Di Egidio]

On Thursday, 31 May 2018 11:57:20 UTC+2, Rupert wrote:
> On Thursday, May 31, 2018 at 11:43:29 AM UTC+2, Julio Di Egidio wrote:

<snip>

> > "infinitary rule
> > of inference" is not even very meaningful.
>
> Finitary rule of inference is set of ordered pairs consisting of a finite set of formulas and a formula. If a pair is in the set then you can infer from the finite collection of formulas which is the first member of the pair, to the formula which is the second member of the pair. Infinitary rule of inference is the same, except that the set of formulas which is the first member of the pair is infinite.

But proofs must be *finite*: there is no such thing as deriving a formula from
an infinite set of premises, not even in infinitary mathematics (where infinite
are the objects, not the proofs)! I am again confused...

Julio

[Rupert]

If you were doing proofs in a system with infinitary rules of inference then you would allow for the possibility that a proof could be a transfinite well-ordered sequence of formulas.

[Julio Di Egidio]

On Thursday, 31 May 2018 14:19:36 UTC+2, Rupert wrote:
> On Thursday, May 31, 2018 at 2:16:40 PM UTC+2, Julio Di Egidio wrote:
> > On Thursday, 31 May 2018 11:57:20 UTC+2, Rupert wrote:
> > > On Thursday, May 31, 2018 at 11:43:29 AM UTC+2, Julio Di Egidio wrote:
> > <snip>
> > > > "infinitary rule
> > > > of inference" is not even very meaningful.
> > >
> > > Finitary rule of inference is set of ordered pairs consisting of a finite set of formulas and a formula. If a pair is in the set then you can infer from the finite collection of formulas which is the first member of the pair, to the formula which is the second member of the pair. Infinitary rule of inference is the same, except that the set of formulas which is the first member of the pair is infinite.
> >
> > But proofs must be *finite*: there is no such thing as deriving a formula from
> > an infinite set of premises, not even in infinitary mathematics (where infinite
> > are the objects, not the proofs)! I am again confused...
>

> If you were doing proofs in a system with infinitary rules of inference then you would allow for the possibility that a proof could be a transfinite well-ordered sequence of formulas.

Which, I'd still contend, doesn't make much sense: proofs *must be finite*, to
be proofs, and those just aren't. (E.g. a proof by induction, even transfinite induction, is still a *finite* proof: in no way it consists in proving the
statement by checking each and every element in the domain. There just is no
such thing as a proof from an infinite set of premises...)

Julio

[Rupert]

Not sure what you're complaining about, really. I'm telling you that when you speak of a "proof" in a system with infinitary rules of inference you relax the requirement that it must be finite. You can call it a "cloof" if you don't want to call it a "proof". I don't see the problem.

[Julio Di Egidio]

On Thursday, 31 May 2018 14:41:31 UTC+2, Rupert wrote:
> On Thursday, May 31, 2018 at 2:29:14 PM UTC+2, Julio Di Egidio wrote:
> > On Thursday, 31 May 2018 14:19:36 UTC+2, Rupert wrote:
> > > On Thursday, May 31, 2018 at 2:16:40 PM UTC+2, Julio Di Egidio wrote:
> > > > On Thursday, 31 May 2018 11:57:20 UTC+2, Rupert wrote:
> > > > > On Thursday, May 31, 2018 at 11:43:29 AM UTC+2, Julio Di Egidio wrote:
> > > > <snip>
> > > > > > "infinitary rule
> > > > > > of inference" is not even very meaningful.
> > > > >
> > > > > Finitary rule of inference is set of ordered pairs consisting of a finite set of formulas and a formula. If a pair is in the set then you can infer from the finite collection of formulas which is the first member of the pair, to the formula which is the second member of the pair. Infinitary rule of inference is the same, except that the set of formulas which is the first member of the pair is infinite.
> > > >
> > > > But proofs must be *finite*: there is no such thing as deriving a formula from
> > > > an infinite set of premises, not even in infinitary mathematics (where infinite
> > > > are the objects, not the proofs)! I am again confused...
> > >
> > > If you were doing proofs in a system with infinitary rules of inference then you would allow for the possibility that a proof could be a transfinite well-ordered sequence of formulas.
> >
> > Which, I'd still contend, doesn't make much sense: proofs *must be finite*, to
> > be proofs, and those just aren't. (E.g. a proof by induction, even transfinite induction, is still a *finite* proof: in no way it consists in proving the
> > statement by checking each and every element in the domain. There just is no
> > such thing as a proof from an infinite set of premises...)
>

> Not sure what you're complaining about, really. I'm telling you that when you speak of a "proof" in a system with infinitary rules of inference you relax the requirement that it must be finite. You can call it a "cloof" if you don't want to call it a "proof". I don't see the problem.

No, you cannot relax that requirement! Of course, the primary problem is
not the misnomer (and that I have never seen it before), but that you are
not appreciating that it's *at best* a misnomer...

Julio

[Peter Percival]

Khong Dong wrote:


> Sure. Anyway you're still unable to answer the above question on how you could
> go from all the terms 0, S0, SS0, ... satisfying P to _all_ individuals
> satisfying P.

In what model are those individuals? If it's a model in which every
individual is named by one of these terms: 0, S0, SS0, ..., then the
inference is valid.


--
Principle of tolerance: it is not our business to set up prohibitions,
but to arrive at conventions. [...] In logic there are no morals.
Everyone is at liberty to build up his own logic, i.e. his own form of
language, as he wishes. -- Rudolph Carnap, /Logical syntax of language/.

[George Greene]

On Wednesday, May 30, 2018 at 1:42:34 AM UTC-4, Khong Dong wrote:
> Fwiw, I've posted (and answered) the question in MSE:
>

> We'll see how that goes I'd guess.

"I guess", not "I'd guess".
Also, "is invalid mathematical reasoning", NOT
"is an invalid mathematical reasoning".
If you can't be bothered to learn how TO USE ARTICLES then how
are you going to be studious enough to learn logic?

"How that goes" is that the question is rated -2 and nobody can
be bothered with it because it's too stupid. In case you didn't
click the link.

You don't actually know anything about "meta-mathematical reasoning".
If you did, you would know that nobody in MSE gives a shit about the
kinds of "meta-mathematical laws" you are trying to introduce there.
They are way too philosophical AND BASIC for any actual mathematicians
to bother with.

[Khong Dong]

On Thursday, 31 May 2018 17:54:42 UTC-6, George Greene wrote:
> On Wednesday, May 30, 2018 at 1:42:34 AM UTC-4, Khong Dong wrote:
> > Fwiw, I've posted (and answered) the question in MSE:
> >
> > We'll see how that goes I'd guess.
>
> "I guess", not "I'd guess".
> Also, "is invalid mathematical reasoning", NOT
> "is an invalid mathematical reasoning".
> If you can't be bothered to learn how TO USE ARTICLES then how
> are you going to be studious enough to learn logic?

I got a good excuse: my ex-professor (and ex-Chairman-of-English-Department)
who retires now told me when my paper goes "live" he'd help fixing the English
part of it as he used to help even math students in the past.

> "How that goes" is that the question is rated -2 and nobody can
> be bothered with it because it's too stupid. In case you didn't
> click the link.

Only "-2"? I even got -googleplex from you, but so what?

> You don't actually know anything about "meta-mathematical reasoning".
> If you did, you would know that nobody in MSE gives a shit about the
> kinds of "meta-mathematical laws" you are trying to introduce there.
> They are way too philosophical AND BASIC for any actual mathematicians
> to bother with.

Sure. I'm sure Godel's and your subjective belief in omega consistency is
million times worst than that: it's a delusion.

[Khong Dong]

On Thursday, 31 May 2018 07:22:44 UTC-6, Peter Percival wrote:
> Khong Dong wrote:
>
>
> > Sure. Anyway you're still unable to answer the above question on how you could
> > go from all the terms 0, S0, SS0, ... satisfying P to _all_ individuals
> > satisfying P.
>
> In what model are those individuals?

No need for the word "model: the individuals are the (individual) natural
numbers. But ...

> If it's a model in which every
> individual is named by one of these terms: 0, S0, SS0, ..., then the
> inference is valid.

The omega consistency stipulation has this part (P(0) and P(S0) and ...), which
though firstordernonrizeable comprises of countably many FOL language terms
hence by essence of upward LST would be inadequate, invalid to ascertain that
it is indeed Ax[P(x)], or not. So omega consistency stipulation is an invalid
one.

[Khong Dong]

Not "firstordernonrizeable" but "nonfirstorderizeable", iirc from TF's posting.

[Khong Dong]

Really? So what has happened to the _concept of the natural numbers_ in you?

Must Godel's paper have depended on the words "standard model" and "non
standard model" before he finished writing his paper?

[Rupert]

None of this has anything to do with Goedel's paper.

[Khong Dong]

So why "standard model" and "non standard model" on the issue of questionable
validity of Godel's omega consistency he wrote in 1931 in my writing in the MSE
thread (and here)?

[Rupert]

You attributed to me the belief that the \omega-rule is a valid rule of inference. So I had to clarify that it was valid for the standard model but not for non-standard models.

As remarked, none of this has anything to do with Goedel's paper. \Omega-consistency, as I actually already tried to explain to you, is the property that a theory T has when you never have any formula with one free variable P(n) such that P(0), P(S0), P(SS0), ... ~(An)P(n), are theorems of the theory T. That's what \omega-consistency is. Nothing you have written has any relevance to what Goedel wrote at all.

[Khong Dong]

Huh? Where did I use the words "valid rule of inference" in conjuntion with
this omega argument.

>
> As remarked, none of this has anything to do with Goedel's paper. \Omega-consistency, as I actually already tried to explain to you, is the property that a theory T has when you never have any formula with one free variable P(n) such that P(0), P(S0), P(SS0), ... ~(An)P(n), are theorems of the theory T. That's what \omega-consistency is. Nothing you have written has any relevance to what Goedel wrote at all.

So how do we know the Godel's sentence G(T) is ... _TRUE_ ?

[Khong Dong]

If all of GIT1 were about purely syntactical provability (theorems) where would
the word "TRUE" come from? From some syntactical rules of inference? And why?

[Rupert]

By GIT1 you presumably mean Proposition VI. The observation that the Goedel sentence is true doesn't come until a fair bit later in the paper. If you wanted to frame the result in a syntactic way you could say that the Goedel sentence for a theory T (satisfying a certain bunch of hypotheses), is provably equivalent in a weak subtheory of T to the assertion that T is consistent. That would be a purely syntactic way of putting the point. Looking quickly at the paper, actually the word "true" does not occur after Section 1. In Proposition X, he speaks of the "validity" of sentences of the form (Ax)F(x) where F is a primitive recursive unary predicate, by which he means "truth". Proposition X shows that given any such sentence its truth is equivalent to the satisfiability of some fixed formula of the restricted predicate calculus. However, Propositions VI and XI, which are usually considered the first and second incompleteness theorem, do not mention truth at all.

[Khong Dong]

I think you had a typo: P(0), P(S0), P(SS0), ... ~(An)P(n) signifies omega-INconsistency.

[Rupert]

What you wrote is correct, and what I wrote is correct as well.

[Peter Percival]

Khong Dong wrote:
> On Thursday, 31 May 2018 23:39:41 UTC-6, Khong Dong wrote:
>> On Thursday, 31 May 2018 07:22:44 UTC-6, Peter Percival wrote:
>>> Khong Dong wrote:
>>>
>>>
>>>> Sure. Anyway you're still unable to answer the above question on how you could
>>>> go from all the terms 0, S0, SS0, ... satisfying P to _all_ individuals
>>>> satisfying P.
>>>
>>> In what model are those individuals?
>>
>> No need for the word "model: the individuals are the (individual) natural
>> numbers. But ...
>>
>>> If it's a model in which every
>>> individual is named by one of these terms: 0, S0, SS0, ..., then the
>>> inference is valid.
>>
>> The omega consistency stipulation has this part (P(0) and P(S0) and ...), which
>> though firstordernonrizeable comprises of countably many FOL language terms
>> hence by essence of upward LST would be inadequate, invalid to ascertain that

No need for the word "model", but LST is all about models. Strange. In
any case one may certainly talk about the standard model without
violating LST.

>> it is indeed Ax[P(x)], or not. So omega consistency stipulation is an invalid
>> one.
>
> Not "firstordernonrizeable" but "nonfirstorderizeable", iirc from TF's posting.
>

[Peter Percival]

Khong Dong wrote:
> Fwiw, I've posted (and answered) the question in MSE:
>
> https://math.stackexchange.com/questions/2801432/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
>
> We'll see how that goes I'd guess.

And where did it go? Do tell.

[Peter Percival]

Khong Dong wrote:


> Must Godel's paper have depended on the words "standard model" and "non
> standard model" before he finished writing his paper?

There is a common misunderstanding about Gödel's incompleteness theorem.
It is often (and wrongly) thought to say that there are statements of
arithmetic which are true but not provable, or not provable in some
class of theories. But it doesn't say that. It says that for theories
T of a rather broad class, there is a statement G (a different G for
each T) such that T neither proves G nor ~G. Models get no mention and
need no mention. In specifying that broad class Gödel used
(unnecessarily, as it happen) the concept of omega consistency. That is
a syntactic matter and, again, models get no and need no mention.

But back to the spurious statement of Gödel's incompleteness theorem -
that there are statements of arithmetic which are true but not provable
in some class of theories. If a statement is not a theorem then it must
be false in some models of the theory (by the completeness theorem). So
what does it mean to say that such a statement is true? It can only
mean that it is true of "the usual arithmetic", the arithmetic we all
learned at school. And the model of that arithmetic is called "the
standard model" it is also called "the intended model".

[Peter Percival]

George Greene wrote:
> On Wednesday, May 30, 2018 at 1:42:34 AM UTC-4, Khong Dong wrote:
>> Fwiw, I've posted (and answered) the question in MSE:
>>
>> We'll see how that goes I'd guess.
>
> "I guess", not "I'd guess".
> Also, "is invalid mathematical reasoning",

Which produces "Is Gödel's Incompleteness Theorem invalid mathematical
reasoning?" for the title, which I think is still wrong. It should be
"Is Gödel's Incompleteness Theorem arrived at by invalid mathematical
reasoning?" Or just "Is Gödel's Incompleteness Theorem invalid?"

> NOT
> "is an invalid mathematical reasoning".
> If you can't be bothered to learn how TO USE ARTICLES then how
> are you going to be studious enough to learn logic?
>
> "How that goes" is that the question is rated -2 and nobody can
> be bothered with it because it's too stupid. In case you didn't
> click the link.
>
> You don't actually know anything about "meta-mathematical reasoning".
> If you did, you would know that nobody in MSE gives a shit about the
> kinds of "meta-mathematical laws" you are trying to introduce there.
> They are way too philosophical AND BASIC for any actual mathematicians
> to bother with.
>
>

[Khong Dong]

Ok. Just two sides of the coin we were talking about.

Any rate, per MSE's suggestion, I've moved the thread there to MO
(Math Overflow), in where the 2nd version is better in elaboration on the
invalidity matter:

https://mathoverflow.net/questions/301610/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning

[Khong Dong]

On Friday, 1 June 2018 04:11:18 UTC-6, Peter Percival wrote:
> Khong Dong wrote:
> > On Thursday, 31 May 2018 23:39:41 UTC-6, Khong Dong wrote:
> >> On Thursday, 31 May 2018 07:22:44 UTC-6, Peter Percival wrote:
> >>> Khong Dong wrote:
> >>>
> >>>
> >>>> Sure. Anyway you're still unable to answer the above question on how you could
> >>>> go from all the terms 0, S0, SS0, ... satisfying P to _all_ individuals
> >>>> satisfying P.
> >>>
> >>> In what model are those individuals?
> >>
> >> No need for the word "model: the individuals are the (individual) natural
> >> numbers. But ...
> >>
> >>> If it's a model in which every
> >>> individual is named by one of these terms: 0, S0, SS0, ..., then the
> >>> inference is valid.
> >>
> >> The omega consistency stipulation has this part (P(0) and P(S0) and ...), which
> >> though firstordernonrizeable comprises of countably many FOL language terms
> >> hence by essence of upward LST would be inadequate, invalid to ascertain that
>
> No need for the word "model", but LST is all about models. Strange. In
> any case one may certainly talk about the standard model without
> violating LST.

The essence of LST (upward) counts: you can prove for instance that essence
using some successor function using some rudimentary intuitive knowledge about
"sets" or Euclidean 2 dimensional points (for its domain, range). That's all
you would need here; you don't need a full blown language structure which a
model of a theory would be.

Go for the essence, not pedantry.

[Khong Dong]

On Friday, 1 June 2018 06:24:03 UTC-6, Peter Percival wrote:
> George Greene wrote:
> > On Wednesday, May 30, 2018 at 1:42:34 AM UTC-4, Khong Dong wrote:
> >> Fwiw, I've posted (and answered) the question in MSE:
> >>
> >> We'll see how that goes I'd guess.
> >
> > "I guess", not "I'd guess".
> > Also, "is invalid mathematical reasoning",
>
> Which produces "Is Gödel's Incompleteness Theorem invalid mathematical
> reasoning?" for the title, which I think is still wrong. It should be
> "Is Gödel's Incompleteness Theorem arrived at by invalid mathematical
> reasoning?" Or just "Is Gödel's Incompleteness Theorem invalid?"

Go for the essence, not pedantry.

[Rupert]

Right. Well, the main thing to be said here is that, as before, your definition of \omega-consistency is completely wrong.

[Peter Percival]

Khong Dong wrote:
> On Friday, 1 June 2018 04:11:18 UTC-6, Peter Percival wrote:
>> Khong Dong wrote:
>>> On Thursday, 31 May 2018 23:39:41 UTC-6, Khong Dong wrote:
>>>> On Thursday, 31 May 2018 07:22:44 UTC-6, Peter Percival wrote:
>>>>> Khong Dong wrote:
>>>>>
>>>>>
>>>>>> Sure. Anyway you're still unable to answer the above question on how you could
>>>>>> go from all the terms 0, S0, SS0, ... satisfying P to _all_ individuals
>>>>>> satisfying P.
>>>>>
>>>>> In what model are those individuals?
>>>>
>>>> No need for the word "model: the individuals are the (individual) natural
>>>> numbers. But ...
>>>>
>>>>> If it's a model in which every
>>>>> individual is named by one of these terms: 0, S0, SS0, ..., then the
>>>>> inference is valid.
>>>>
>>>> The omega consistency stipulation has this part (P(0) and P(S0) and ...), which
>>>> though firstordernonrizeable comprises of countably many FOL language terms
>>>> hence by essence of upward LST would be inadequate, invalid to ascertain that
>>
>> No need for the word "model", but LST is all about models. Strange. In
>> any case one may certainly talk about the standard model without
>> violating LST.
>
> The essence of LST (upward) counts: you can prove for instance that essence
> using some successor function using some rudimentary intuitive knowledge about
> "sets" or Euclidean 2 dimensional points (for its domain, range

You've lost me. Domain and range of what?

> ). That's all
> you would need here; you don't need a full blown language structure which a
> model of a theory would be.
>
> Go for the essence, not pedantry.
>
>
>>
>>>> it is indeed Ax[P(x)], or not. So omega consistency stipulation is an invalid
>>>> one.
>>>
>>> Not "firstordernonrizeable" but "nonfirstorderizeable", iirc from TF's posting.

[Peter Percival]

Khong Dong wrote:


> Any rate, per MSE's suggestion, I've moved the thread there to MO
> (Math Overflow), in where the 2nd version is better in elaboration on the
> invalidity matter:
>
> https://mathoverflow.net/questions/301610/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
>

You still use |= when discussing omega consistency. It should be |-.
Omega consistency is a syntactic matter.

[Peter Percival]

If "pedantry" is just a rude word for correctness, then I'd much rather
go for correctness. But what do you think is the "essence" of upward LST?

>>>> it is indeed Ax[P(x)], or not. So omega consistency stipulation is an invalid
>>>> one.
>>>
>>> Not "firstordernonrizeable" but "nonfirstorderizeable", iirc from TF's posting.

[Khong Dong]

On Friday, 1 June 2018 09:11:13 UTC-6, Peter Percival wrote:
> Khong Dong wrote:
>
>
> > Any rate, per MSE's suggestion, I've moved the thread there to MO
> > (Math Overflow), in where the 2nd version is better in elaboration on the
> > invalidity matter:
> >
> > https://mathoverflow.net/questions/301610/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
> >
>
> You still use |= when discussing omega consistency. It should be |-.
> Omega consistency is a syntactic matter.

*Only* syntactical matter? You're still wrong.

[Peter Percival]

Perhaps you could write down what you think "T is omega consistent"
means, then we can see if there's anything beyond syntax in it.

[Khong Dong]

On Friday, 1 June 2018 13:05:55 UTC-6, Peter Percival wrote:
> Khong Dong wrote:
> > On Friday, 1 June 2018 09:11:13 UTC-6, Peter Percival wrote:
> >> Khong Dong wrote:
> >>
> >>
> >>> Any rate, per MSE's suggestion, I've moved the thread there to MO
> >>> (Math Overflow), in where the 2nd version is better in elaboration on the
> >>> invalidity matter:
> >>>
> >>> https://mathoverflow.net/questions/301610/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
> >>>
> >>
> >> You still use |= when discussing omega consistency. It should be |-.
> >> Omega consistency is a syntactic matter.
> >
> > *Only* syntactical matter? You're still wrong.
> >
> Perhaps you could write down what you think "T is omega consistent"
> means, then we can see if there's anything beyond syntax in it.

Why don't you go reading my MO post.

[Peter Percival]

I have, and (as remarked) you use |= where you should have |-. This is
what I think omega consistent means. (If Rupert says otherwise, I'll
defer to him.)

A theory that interprets arithmetic is omega consistent if this is *not*
the case: there is some formula phi(x) such that the theory proves
phi(n) for each numeral n denoting a standard natural number and it also
proves (Ex)(~phi(x)).

Note that that is about proof, a syntactic notion.

[Khong Dong]

No. You're mistaken: you were just unable to understand the issue.

[Khong Dong]

On Friday, 1 June 2018 13:56:21 UTC-6, Peter Percival wrote:
> Khong Dong wrote:
> > On Friday, 1 June 2018 13:05:55 UTC-6, Peter Percival wrote:
> >> Khong Dong wrote:
> >>> On Friday, 1 June 2018 09:11:13 UTC-6, Peter Percival wrote:
> >>>> Khong Dong wrote:
> >>>>
> >>>>
> >>>>> Any rate, per MSE's suggestion, I've moved the thread there to MO
> >>>>> (Math Overflow), in where the 2nd version is better in elaboration on the
> >>>>> invalidity matter:
> >>>>>
> >>>>> https://mathoverflow.net/questions/301610/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
> >>>>>
> >>>>
> >>>> You still use |= when discussing omega consistency. It should be |-.
> >>>> Omega consistency is a syntactic matter.
> >>>
> >>> *Only* syntactical matter? You're still wrong.
> >>>
> >> Perhaps you could write down what you think "T is omega consistent"
> >> means, then we can see if there's anything beyond syntax in it.
> >
> > Why don't you go reading my MO post.
>
> I have, and (as remarked) you use |= where you should have |-. This is
> what I think omega consistent means. (If Rupert says otherwise, I'll
> defer to him.)
>
> A theory that interprets arithmetic is omega consistent if this is *not*
> the case: there is some formula phi(x) such that the theory proves
> phi(n) for each numeral n denoting a standard natural number and it also
> proves (Ex)(~phi(x)).
>
> Note that that is about proof, a syntactic notion.

Then write down the consistency down as a first and we perhaps can talk.

[Peter Percival]

You really, really, really ought to know what it means to say that a
theory is consistent without anyone having to tell you, but if you
insist... A theory T is said to be consistent if there is some formula
in the language of the theory that the theory does not prove.

[Khong Dong]

Oh sorry for the typo: I meant write down omega consistent T.

[Peter Percival]

Khong Dong wrote:
> [...] write down omega consistent T.
>
T is omega consistent if there is no formula phi(x) such that the theory
proves phi(n) for each numeral n denoting a standard natural number, and
it also proves ~(Ux)(phi(x)).

That is how Gödel defined it in /On formally undecidable.../.

[Khong Dong]

On Friday, 1 June 2018 16:58:38 UTC-6, Peter Percival wrote:
> Khong Dong wrote:
> > [...] write down omega consistent T.
> >
> T is omega consistent if there is no formula phi(x) such that the theory
> proves phi(n) for each numeral n denoting a standard natural number, and
> it also proves ~(Ux)(phi(x)).
>
> That is how Gödel defined it in /On formally undecidable.../.

On which page did Gödel define "standard natural number"?
On which page did Gödel define "non standard natural number"?

[Peter Percival]

Khong Dong wrote:
> On Friday, 1 June 2018 16:58:38 UTC-6, Peter Percival wrote:
>> Khong Dong wrote:
>>> [...] write down omega consistent T.
>>>
>> T is omega consistent if there is no formula phi(x) such that the theory
>> proves phi(n) for each numeral n denoting a standard natural number, and
>> it also proves ~(Ux)(phi(x)).
>>
>> That is how Gödel defined it in /On formally undecidable.../.
>
> On which page did Gödel define "standard natural number"?

The numerals for them are 0, f0, ff0, ...

> On which page did Gödel define "non standard natural number"?

What's that got to do with anything?

[Peter Percival]

Khong Dong wrote:
> On Friday, 1 June 2018 16:58:38 UTC-6, Peter Percival wrote:
>> Khong Dong wrote:
>>> [...] write down omega consistent T.
>>>
>> T is omega consistent if there is no formula phi(x) such that the theory
>> proves phi(n) for each numeral n denoting a standard natural number, and
>> it also proves ~(Ux)(phi(x)).
>>
>> That is how Gödel defined it in /On formally undecidable.../.
>
> On which page did Gödel define "standard natural number"?

Since it's a syntactic matter the numbers get no mention, it's the
numerals that appear in the formula phi. Gödel's definition (using "F"
not phi) spans pages 214 and 215; but I was wrong about the paper, it is
in /Einige metamathematische Resultate.../ of 1930.

> On which page did Gödel define "non standard natural number"?
>

[Rupert]

Gee, you're pretty confident about that, eh?

[Rupert]

On Friday, June 1, 2018 at 9:56:21 PM UTC+2, Peter Percival wrote:
> Khong Dong wrote:
> > On Friday, 1 June 2018 13:05:55 UTC-6, Peter Percival wrote:
> >> Khong Dong wrote:
> >>> On Friday, 1 June 2018 09:11:13 UTC-6, Peter Percival wrote:
> >>>> Khong Dong wrote:
> >>>>
> >>>>
> >>>>> Any rate, per MSE's suggestion, I've moved the thread there to MO
> >>>>> (Math Overflow), in where the 2nd version is better in elaboration on the
> >>>>> invalidity matter:
> >>>>>
> >>>>> https://mathoverflow.net/questions/301610/is-g%C3%B6dels-incompleteness-theorem-an-invalid-meta-mathematical-reasoning
> >>>>>
> >>>>
> >>>> You still use |= when discussing omega consistency. It should be |-.
> >>>> Omega consistency is a syntactic matter.
> >>>
> >>> *Only* syntactical matter? You're still wrong.
> >>>
> >> Perhaps you could write down what you think "T is omega consistent"
> >> means, then we can see if there's anything beyond syntax in it.
> >
> > Why don't you go reading my MO post.
>
> I have, and (as remarked) you use |= where you should have |-. This is
> what I think omega consistent means. (If Rupert says otherwise, I'll
> defer to him.)
>
> A theory that interprets arithmetic is omega consistent if this is *not*
> the case: there is some formula phi(x) such that the theory proves
> phi(n) for each numeral n denoting a standard natural number and it also
> proves (Ex)(~phi(x)).

Yes.

[Khong Dong]

Yes.

[Khong Dong]

On Friday, 1 June 2018 17:23:18 UTC-6, Peter Percival wrote:
> Khong Dong wrote:
> > On Friday, 1 June 2018 16:58:38 UTC-6, Peter Percival wrote:
> >> Khong Dong wrote:
> >>> [...] write down omega consistent T.
> >>>
> >> T is omega consistent if there is no formula phi(x) such that the theory
> >> proves phi(n) for each numeral n denoting a standard natural number, and
> >> it also proves ~(Ux)(phi(x)).
> >>
> >> That is how Gödel defined it in /On formally undecidable.../.
> >
> > On which page did Gödel define "standard natural number"?
>
> Since it's a syntactic matter the numbers get no mention,

What a laugh. _TRUE_ and un-provable is syntactic matter only!

[Khong Dong]

On Friday, 1 June 2018 01:33:49 UTC-6, Rupert wrote:

> > If all of GIT1 were about purely syntactical provability (theorems) where would
> > the word "TRUE" come from? From some syntactical rules of inference? And why?
>
> By GIT1 you presumably mean Proposition VI. The observation that the Goedel
> sentence is true doesn't come until a fair bit later in the paper.

Does it matter where in the paper it comes?

> If you wanted to frame the result in a syntactic way you could say that the
> Goedel sentence for a theory T (satisfying a certain bunch of hypotheses),

"satisfying a certain bunch of hypotheses"? Yeah, like T must convey non syntactical truths of the concept of natural numbers!

[Rupert]

No. Any theory T in which Elementary Function Arithmetic is interpretable and which is \omega-consistent. You do not need arithmetical soundness.

[Rupert]

So, you're pretty confident that you're right, and I'm pretty confident that you're wrong, and you've offered no reasoning at all in support of your view. So where does that leave us?

[Rupert]

The word "true" does not occur in the paper except for Section 1 (and also the word "validity", meaning "truth", occurs in Proposition X). As Peter correctly says, Propositions VI and XI, which are generally reckoned to be the first and second incompleteness theorems, use no semantic notions at all.

[Khong Dong]

Yeah. That's why one has to question your and Peter's competency on understanding Godel's
paper. Godel had to use the word "true" or "truth"? The countless occurrences of "natural
numbers" or that of something like "proposition about natural numbers" wouldn't be enough to
make GIT1 semantic in nature? Wow! Where's your competency?

[Rupert]

Yeah. That's about the size of it.

> Wow! Where's your competency?

Do you think there's some possibility you just over-estimate your competence to evaluate other people's competence?

[Khong Dong]

On this subject of the invalidity of Godel's Omega assumption for the concept of
the natural numbers, No.

[Khong Dong]

That is "Omega Consistency assumption for the concept of the natural numbers".

[Khong Dong]

Ultimately you et al. should realize FOL language can't adequately express the specific size of
an infinity and Godel inadvertently cheated that fact in his paper with his omega consistency
assumption on the informal concept of the natural numbers.

[Khong Dong]

The phrase "the set of natural numbers" is actually a concept variable ranging over some
collection of non-logical concepts. Think about that, Rupert.

[Rupert]

Well, you ought to re-evaluate that opinion of yours.

[Rupert]

Wrong.

[Rupert]

So why exactly should I believe that?

[Khong Dong]

Is one a prime number to you?

[Rupert]

The usual definition of prime number includes that it is required to be greater than one.

[Peter Percival]

I think you're muddling two things: the incompleteness theorem, and
omega consistency.

Here is a quotation of GIT1:

Theorem VI. For every omega-consistent recursive class kappa of FORMULAS
there are recursive CLASS SIGNS r such that neither v Gen r nor Neg(v
Gen r) belong to Flg(kappa) (where v is the FREE VARIABLE of r).

That is from the English translation that may be found in van
Heijenoort's collection and in volume I of Gödel's collected works. It
is wholly syntactic. I don't expect you to be convinced, for who's to
say that the definitions of some of the concepts used in the statement
of the theorem are not defined semantically? To settle that one must
follow the chains of definitions back to the undefined symbols. Am I
going to write the results down here? No, it will be necessary for
_you_ to _read_ the paper. Am I right to suspect that you've never done
that?

There is an accidental side-effect (so to speak) of Thm VI. 'v Gen r'
and 'Neg(v Gen r)' are closed formulae, so one of them is true. But the
theorem doesn't say that. [Here I'm being very sloppy because of course
'v Gen r' and 'Neg(v Gen r)' _aren't_ formulae, rather there are Gödel
numbers of formulae...]

The definition of omega consistency (to be found in the same places) is
that for no F(x) are F(1), F(2), ..., F(n),... ad infinitum and
(Ex)~F(x) all provable. Note that 1, 2, ... are _numerals_ (in Gödel's
P they are f0, ff0, ...), they must be for F(etc) to be formulae.

[Julio Di Egidio]

On Saturday, 2 June 2018 12:25:45 UTC+2, Peter Percival wrote:
> Khong Dong wrote:

<snip>

> > What a laugh. _TRUE_ and un-provable is syntactic matter only!
> >
> I think you're muddling two things: the incompleteness theorem, and
> omega consistency.
>
> Here is a quotation of GIT1:
>
> Theorem VI. For every omega-consistent recursive class kappa of FORMULAS
> there are recursive CLASS SIGNS r such that neither v Gen r nor Neg(v
> Gen r) belong to Flg(kappa) (where v is the FREE VARIABLE of r).
>
> That is from the English translation that may be found in van
> Heijenoort's collection and in volume I of Gödel's collected works. It
> is wholly syntactic.

*That* statement may be fully syntactic, the theorem won't be just because
that statement is, so yours (and Rupert's) are already fallacies anyway and
as usual (and I am not even getting into who is more presumptuous in these
threads).

In fact, not even that much I'd concede: that statement (and the steps to it)
look syntactic, yet the proof predicate ("Bew" or however it's called in the
original) is NOT recursive, as anyway noted by Goedel himself, hence that the
argument is purely syntactic remains a *false myth*. Rather, that *THERE JUST
IS NO SUCH THING as a purely syntactical meta-mathematical argument* I take to
be the actual lesson from that exercise and another name for "incompleteness".

Julio

[Julio Di Egidio]

P.S. In that light it is also clear why G is TRUE and anything else is at least
unsound.

Julio

[Khong Dong]

And the _usual definition_ of prime number to Golbach, Euler, and to other
_mathematicians_ included that one, being less than two, to be a prime.

So do you now concede that the phrase "the set of natural numbers" is actually
a concept variable ranging over some collection of non-logical concepts?

[Rupert]

Can't see the relevance of the point that about prime numbers to what you're saying.

[Khong Dong]

On Saturday, 2 June 2018 10:00:45 UTC-6, Rupert wrote:
> Can't see the relevance of the point that about prime numbers to what you're saying.

Because either you snipped that point or you've been being unable to undersdtand
it. Here's the point again:

the phrase "the set of natural numbers" is actually a concept variable ranging

over some collection of non-logical concepts.

As a competent poster, would you concede/agree that the phrase "the set of

[Khong Dong]

Goedel _had to_ stipulate the concept of the natural numbers be such that the
omega consistency is true of them because otherwise he would have had no ground
to claim G is TRUE.

But since Goedel's omega consistency stipulation is invalid, so is his G(T)
being true. It's that simple but Rupert et al. couldn't understand.


>
> Julio

[Peter Percival]

Khong Dong wrote:


> But since Goedel's omega consistency stipulation is invalid,

What is the problem with omega consistency?

> so is his G(T)
> being true.

You know that the assumption of omega consistency is not necessary for
incompleteness, don't you? The sentence (usually called G here) that is
such that neither it nor its logical negation is provable, is now a
different formula but incompleteness is still provable with the weaker
assumption of consistency.

[Khong Dong]

Depending on whose translation, but on the one by B. Meltzer, pg. 39, Godel
wrote:

"Accordingly, then, a formula is a finite series of natural numbers, and
[...]. Meta-mathematical concepts and propositions thereby become concepts
and propositions concerning natural numbers, or series of them".

So it's more than just formal systems. After all his requirement would also include that the formal system *carry basic concepts of the natural numbers*.

The syntactical parts of the omega consistency stipulation merely _carry_
semantically the omega consistency stipulation on the concept of the natural
numbers. How many times would you and Peter need to be informed, told on this
trivial knowledge?