interpretative ideas forged by Gödel's Incompleteness Theorems, — Aryamoy Mitra

Fastest way: Godel's theorems are rigorous arguments with a rigorously determined subject matter. We understand, then, or not, or to some degree which itself requires gauging. Interpretation, on the other hand, is all the area outside the rigor, and while some fun, even poetically interesting, most is nonsense.

1) Any arithmetic model? No.

2) Crisis at the heart? No.

3) Transcending systems? No. Although one can think about the system as a system, from "outside" of the system. In this instance called meta-mathematical thinking. And I think that one solves all Godelian problems by adding as axioms the critical sentences. In any case, there is no "Royal road to knowledge" certainly with respect to Godel's theories and thinking, there's just the work of it.

Fastest way: Godel's theorems are rigorous arguments with a rigorously determined subject matter. We understand, then, or not, or to some degree which itself requires gauging. Interpretation, on the other hand, is all the area outside the rigor, and while some fun, even poetically interesting, most is nonsense. — tim wood

Interpretations may not be sacrosanct to formalized systems of logic, but are they truly nonsensical? Any epistemological modality that isn't purely formalized, is likely underpinned and inspired by interpretations. Not all of them have to be whimsical, or poetically arbitrary; oftentimes, they're necessary in discerning or directing what neither empiricism, nor formal logic can (eg - hidden variable theories in canonical QM paradigms).

1) Any arithmetic model? No. — tim wood

Thanks - that's what the literature suggests, too.

2) Crisis at the heart? No — tim wood

Once again, a presumed extension of the first answer.

3) Transcending systems? No. Although one can think about the system as a system, from "outside" of the system. In this instance called meta-mathematical thinking. And I think that one solves all Godelian problems by adding as axioms the critical sentences. In any case, there is no "Royal road to knowledge" certainly with respect to Godel's theories and thinking, there's just the work of it. — tim wood

Contemplating a system externally, is what I meant by 'transcending' them.

Isn't there, however, a self-perpetuating element to Gödel incompleteness (that is, irrespective of how many new axioms one defines a set or structure with, an unprovable sentence can always be derived within it)?

Isn't there, however, a self-perpetuating element to Gödel incompleteness (that is, irrespective of how many new axioms one defines a set or structure with, an unprovable sentence can always be derived within it)? — Aryamoy Mitra

I think the idea is that any system of sufficient power is subject to Godelian self-referential sentences that are in that system unprovable, but provable in an expanded system that incudes the sentence as an axiom, that extendibility into the transfinite. But if you run with this, your almost certainly running into speculation, which Godel really is not about. Your choice: riff on it in some ultimately nonsensical way or read a book or two. Godel's 1931 undecidability paper in English is very readable.

Secondly, doesn't the absence of self-consistency foreshadow, that an intractable crisis permeates the heart of all (conceivable) logical architectures? — Aryamoy Mitra

In my view, no.

What Gödel's incompleteness Theorems and other Incompleteness results (Turing, Church) simply show are the limitations of giving a direct proof. And do notice that with giving an indirect proof you cannot say so much as with a direct proof.

Notice that something existing (being logical correct in this case) and something being provable are two different things. This is the basic underlying issue here.

systematically finite — Aryamoy Mitra

characterized by Peano Arithmetic — Aryamoy Mitra

simultaneously unprovable — Aryamoy Mitra

entirely bereft of composite statements — Aryamoy Mitra

ceases to unequivocally demonstrate its own mathematical consistency — Aryamoy Mitra

Where do you find such terminology in discussions of incompleteness? Where did you read such things?

Meanwhile, it's better to look at Godel-Rosser incompleteness, since it is stronger than Godel incompleteness and is what most people mean by now when they refer to incompleteness. Also, we should take advantage of certain refinements in mathematical logic that were not present when Godel proved incompleteness.

Godel-Rosser is that any recursively axiomatizable and consistent theory that "expresses enough arithmetic" (such as Robinson arithmetic or Peano arithmetic) is incomplete in the sense that there are sentences in the language of theory such that neither the sentence nor its negation is a theorem of the theory.

From the above it follows that for such theories, there are true (in this context meaning true in the standard model of PA) sentences that are not theorems. Indeed, we may construct a specific such sentence.

Moreover, no such system proves its own consistency.

/

Godel-Rosser has proof that is constructive, intuitionistically (cf. philosophy of intuitionism in mathematics) acceptable, and finitistic (the proof can be carried out in primitive recursive arithmetic).

/

when accorded a finite list of axioms or postulates, do there always result either indeterminate, or unprovable truth values — Aryamoy Mitra

If a theory is recursively axiomatizable, sufficiently arithmetically expressive, and consistent (let's call these 'G-theories'), then it is incomplete, no matter whether the set of axioms is finite or infinite. Some theories that are not G- theories are complete (and some are finitely axiomatizable).

absence of self-consistency — Aryamoy Mitra

What lack of consistency? Incompleteness doesn't say that e.g. PA or ZFC are inconsistent. Rather, a proof of consistency is not available within their own systems.

finite set of elementary axioms — Aryamoy Mitra

Some theories are recursively axiomatizable with even an infinite set of axioms. For some reason you are stuck on a notion of finite axiomatization that is not relevant in this regard.

Also, 'elementary' has a technical meaning different from your use.

in order to illustrate a logical system's consistency, one must transcend the system entirely — Aryamoy Mitra

You shouldn't generalize about "logical systems" but rather you should be accurate by addressing just G-theories in this context. And the answer is yes; to prove the consistency of a G-theory we have to do that in some other theory (which itself could be another G-theory). For example, Z set theory proves the consistency of PA.

Isn't that tantamount, for instance, to asserting that all finitely synthesized constructs of reasoning, are by their existence inconsistent? — Aryamoy Mitra

Not at all. The question itself shows a confused understanding of this topic. If you are interested in understanding incompleteness, I suggest studying it from good sources.

how many new axioms one defines a set or structure with, an unprovable sentence can always be derived within it — Aryamoy Mitra

The wording of that is not good, but basically yes, you can't add axioms to a G-theory to escape incompleteness.

Where do you find such terminology in discussions of incompleteness? Where did you read such things? — TonesInDeepFreeze

Neither of them are terminological, in the first place.

If a theory is recursively axiomatizable, sufficiently arithmetically expressive, and consistent (let's call these 'G-theories'), then it is incomplete, no matter whether the set of axioms is finite or infinite. Some theories that are not G- theories are complete (and some are finitely axiomatizable). — TonesInDeepFreeze

Here's the crux of the question I was posturing - insofar as arithmetic expression and consistency are prerequisites to Gödel incompleteness.

What lack of consistency? Incompleteness doesn't say that e.g. PA or ZFC are inconsistent. Rather, a proof of consistency is not available within their own systems. — TonesInDeepFreeze

That is literally, what 'self-consistency' denotes (demonstrating a self-referential consistency, from within a system).

Some theories are recursively axiomatizable with even an infinite set of axioms. For some reason you are stuck on a notion of finite axiomatization that is not relevant in this regard. — TonesInDeepFreeze

There's no fixation on 'finite axiomatization' - I'm only asking whether given a condition of axiomatic finiteness, Gödel incompleteness extends outside Peano Arithmetic.

Also, 'elementary' has a technical meaning different from your use. — TonesInDeepFreeze

What are the (formal) subtleties?. What does a non-elementary axiom entail?

You shouldn't generalize about "logical systems" but rather you should be accurate by addressing just G-theories in this context. And the answer is yes; to prove the consistency of a G-theory we have to do that in some other theory (which itself could be another G-theory). For example, Z set theory proves the consistency of PA. — TonesInDeepFreeze

Discerning which generalizations are appropriate, and which aren't, was the predominant objective of this thread. G-theories, as I'm sure you'd concur, are not a trivial subset of all logical formalisms. If, for example, one were to create and quantify a logical edifice (with arithmetic), might the Gödelian constraints on certain proof-statement correspondences in formal languages, lend itself to the underlying logical edifice?

I think the idea is that any system of sufficient power is subject to Godelian self-referential sentences that are in that system unprovable, but provable in an expanded system that incudes the sentence as an axiom, that extendibility into the transfinite. But if you run with this, your almost certainly running into speculation, which Godel really is not about. — tim wood

Why is it speculative, though?

Neither of them are terminological, in the first place. — Aryamoy Mitra

Sincerely, I would like to help you understand this topic and to provide answers, but at many points I don't know what you're trying to say because you use unrecognizable or too vague terminology.

What lack of consistency? Incompleteness doesn't say that e.g. PA or ZFC are inconsistent. Rather, a proof of consistency is not available within their own systems.
— TonesInDeepFreeze

That is literally, what 'self-consistency' denotes (demonstrating a self-referential consistency, from within a system). — Aryamoy Mitra

'self-consistency' is not ordinarily used in the sense of "proves its own consistency". Rather, 'self-consistency' is just a longer phrase for 'consistency'.

A theory is consistent if and only if there is not a formula such that the theory proves both the formula and the negation of the formula.

A theory T proves the consistency of a theory S if and only if T proves that that S does not prove a formula and its negation. In particular, a theory T proves the consistency of T if and only if T proves that T does not prove a formula and its negation.

A theory may be consistent and not prove its own consistency.

axiomatic finiteness — Aryamoy Mitra

Do you mean that there are only a finite number of axioms? Or do you mean that the axioms entail that there are only finite sets, or something like that?

whether [...] Gödel incompleteness extends outside Peano Arithmetic. — Aryamoy Mitra

What do you mean by 'extends outside' a theory?

Do you mean to ask whether there are theories stronger than PA that are incomplete. Yes.

Or theories with all the axioms of PA plus more axioms and that are incomplete? Yes.

What does a non-elementary axiom entail? — Aryamoy Mitra

Often, 'elementary' means first-order. Or, in different contexts, it refers to elementary arithmetic with elementary functions, which have a specific technical definition as a certain subset of the set of recursive functions.

might the Gödelian constraints on certain proof-statement correspondences in formal languages, lend itself to the underlying logical edifice? — Aryamoy Mitra

I don't know what you mean by 'proof-statement correspondences' nor what you mean by 'underlying logical edifice'.

Sincerely I say that your understanding of this subject would depend on familiarizing yourself with good books or articles on it, and with that you would have recognizable terminology in which to couch your questions about it.

Highly recommended:

'Godel's Theorem: An Incomplete Guide To Its Use And Abuse' - Torkel Franzen

Probably the best book ever written for introducing the subject of incompleteness to everday readers.

'self-consistency' is not ordinarily used in the sense of "proves its own consistency". Rather, 'self-consistency' is just a longer phrase for 'consistency'.

A theory is consistent if and only if there is not a formula such that the theory proves both the formula and the negation of the formula. — TonesInDeepFreeze

I was merely shortening (with 'self-consistency'), how I communicated the notion of a (system) proving its own consistency; it may not have been of appropriacy.

Do you mean that there are only a finite number of axioms? Or do you mean that the axioms entail that there are only finite sets, or something like that? — TonesInDeepFreeze

- a finite number of axioms.

Do you mean to ask whether there are theories stronger than PA that are incomplete. Yes.

Or theories with all the axioms of PA plus more axioms and that are incomplete? Yes. — TonesInDeepFreeze

Both; as long as it doesn't consist solely of PA. Thank you, for clarifying that the answer is affirmative either way.

I don't know what you mean by 'proof-statement correspondences' nor what you mean by 'underlying logical edifice'. — TonesInDeepFreeze

First-order logic (unless I'm mistaken) is a corollary of propositional logic, insofar as it quantifies the interrelations between its subjects - as opposed to delineating them with logical connectives. With a 'logical edifice', I was referring to a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators. Won't any constraints on the latter, inclusive of Gödel incompleteness, emerge for the former (propositional ideas)? Is this formulation, any more intelligible?

Sincerely I say that your understanding of this subject would depend on familiarizing yourself with good books or articles on it, and with that you would have recognizable terminology in which to couch your questions about it. — TonesInDeepFreeze

'Godel's Theorem: An Incomplete Guide To Its Use And Abuse' - Torkel Franzen
Probably the best book ever written for introducing the subject of incompleteness for everday readers. — TonesInDeepFreeze

That's most certainly an endeavor of mine. I'm not as educated on this subject as you are, since propositional calculus (and its formalized language) is a rarefied discipline. Hopefully, nonetheless, I'll one day be able to contextualize Gödel's incompleteness theorems - as they are meant to be.

My bad. I wasn't clear. There are lots of people who think Godel's undecidable theorems are applicable to practically anything they can think of - and they're not. Theirs just speculation based on an essential ignorance of the the subject. Going that way is what I meant by speculation. The subject itself, though, is very, very dry. By all means consult secondary literature. But if you can find Goldel's 1931 paper in English, it's a pleasure to read, and a person with not too much work can slog through it. It's not the language of it, but the ideas, and not even so much that, but how they're put together.

a finite number of axioms — Aryamoy Mitra

Then I don't know what relevance you have in mind. G-theories can have finitely many or infinitely many axioms.

First-order logic (unless I'm mistaken) is a corollary of propositional logic — Aryamoy Mitra

No, the opposite. First-order logic subsumes propositional logic.

[First-order logic] quantifies the interrelations between its subjects - as opposed to delineating them with logical connectives. — Aryamoy Mitra

First order logic allows predicate symbols, operation symbols, and quantifers, which are not present in propositional logic. But first-order logic does have the connectives also.

With a 'logical edifice', I was referring to a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators. Won't any constraints on the latter, inclusive of Gödel incompleteness, emerge for the former (propositional ideas)? — Aryamoy Mitra

I cannot make sense of that. I don't know what you mean by "a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators".

No, the opposite. First-order logic subsumes propositional logic. — TonesInDeepFreeze

Yes, that's what I meant with the term 'corollary'. First-order logic, in most contexts, cannot exist without an underlying propositional logic (once again, unless I'm mistaken).

I cannot make sense of that. I don't know what you mean by "a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators". — TonesInDeepFreeze

If logical propositions can be quantified with quantifiers, then can they not (equivalently) be mediated with arithmetic operators (that is to say, 'adding' and 'subtracting' propositional conditions from one another)?

First-order logic, in most contexts, cannot exist without an underlying propositional logic (once again, unless I'm mistaken). — Aryamoy Mitra

You are correct.

'adding' and 'subtracting' propositional conditions from one another — Aryamoy Mitra

I don't know what you mean.

I've been trying to wrap my head around this for a long time and though I've read it being stated clearly as true I still don't understand how something can be "true" and "unprovable". Truth has to be established i.e. it has to be proven and if something is "true" then necessarily that it's been "proven". Then the Godel sentence becomes, "proven" (as true) AND "unprovable". Isn't this a contradiction?

I still don't understand how something can be "true" and "unprovable" — TheMadFool

This is explained in any textbook in mathematical logic, usually chapters 1 and 2.

Proof concerns just formulas in the language - purely syntactical objects.

Truth concerns models for the language. A sentence of the language might be true in some models and false in other models. With incompleteness, by 'true' we leave tacit that we actually mean true in one particular model, which is the standard model for the language of PA.

Proof takes place merely with regard to the theory itself. Truth is handled by a meta-theory (usually set theory) for the theory.

A theorem of a theory is a sentence that is provable from the axioms for the theory.

We say a model is a model of a given theory if and only if every theorem of the theory is true in the model.

If a sentence is provable from the axioms of a given theory, then that sentence is true in all models of the theory.

But a sentence might be true in a given model of the theory and yet not be provable from the axioms of the theory.

Incompleteness of a theory is that there are sentences such that neither the sentence nor its negation is provable from axioms for the theory. But in any given model, a sentence is either true or false. So for an incomplete theory, there are sentences that are true in certain models but not provable. In particular, for certain theories, we show that there are unprovable sentences that are true in the standard model for the language of PA.

1. The theorem F1 is true AND the theorem F1 unprovable (Assume for reductio ad absurdum)

2. IF the theorem F1 is unprovable THEN we don't know the truth value of theorem F1

3 The theorem F1 is true (1 Simp)

4. IF the theorem F1 is true THEN we know the truth value of theorem F1

5. We know the truth value of theorem F1 (3, 4 Modus Ponens)

6. The theorem F1 is unprovable (1 Simp)

7. We don't know the truth value of theorem F1 (2, 6 Modus ponens)

8. We know the truth value of theorem F1 AND We don't know the truth value of thoerem F1 (5, 7 Conj)

9. False that the theorem F1 is true AND the theorem F1 is unprovable (1 to 8 reductio ad absurdum)

:chin:

There are lots of people who think Godel's undecidable theorems are applicable to practically anything they can think of - and they're not. — tim wood

Incompleteness theorems.

Even if there are the obvious nonsense (just think how much nonsense there is about quantum physics etc), I think there are vast more of those mathematicians who push the theorem to the sidelines close to the border of logic and logical inquiry and insist that it has nothing to do with anything else in the field of math than what the theorems state. It's just an oddity and no reason to think just what those Gödel numbers would be. I think many of these don't even see any link to Turing's Halting problem or with other incompleteness results.

Here's a simpified synopsis of the terminology:

SYNTACTICAL:

1. We have formal languages. These are sets of symbols. And there are rules for sequencing the symbols to make terms and formulas. Sentences are a certain kind of formula. It is computer-checkable whether a given sequence of symbols is or is not a formula for a given formal language. And it is computer-checkable whether a given sequence of symbols is or is not a formula of a given formal language.

2 We have rules of proof. A proof is a certain kind of sequence of formulas. It is computer-checkable whether a given sequence of formulas is or is not a proof.

3. A theory is a set of sentences closed under proof. A theorem is a member of that set of sentences.

4. An axiomatization for a theory is a set of formulas such that every member of the theory is provable from the axioms. So a theorem is a sentence that is provable from a set of axioms for the theory.

4. A theory is decidable if and only if it is computer-checkable whether any given sentence is or is not a member of the theory.

5. A theory is complete if and only if for any given sentence in the language, either the sentence or its negation is a member of the theory.

6. A theory is consistent if and only if there is no sentence such that both the sentence and its negation is a member of the theory.

7. A theory is recursively axiomatizable if and only if there there is a computer-checkable set of axioms for the theory.

8. The notion of a theory being "sufficiently arithmetically expressive" is complicated and can't be summarized here. But basically it means that the theory has the ordinary basic truths about arithmetic.

9. Godel-Rosser incompleteness is that there are recursively axiomatized, consistent, sufficiently arithmetically expressive theories that are incomplete.

P.S. Note that when we say 'unprovable' we mean 'unprovable from certain axioms'. No formula is absolutely unprovable, since any formula is provable from certain axioms (even if just from an inconsistent set of axioms).

SEMANTIC:

10. A model for a language is a non-empty set that is the universe for the model and a mapping from the symbols of the language to individuals in the set, relations on the set, and functions on the set.

11. A model then describes a "state of affairs". That is, some members of the universe and tuples of members are either in the relations or not, or in the functions or not.

12. The Tarski method is applied so that 'truth in the model' is built up in stages for simple sentences to more complicated sentences. A sentence is either true in a given model or false in that model.

13. PA, in this context, is a certain first-order theory.

14. By 'the standard model for the language of PA' we mean the model where the symbols of the language map to the natural numbers and functions on the natural numbers in the way we would expect. For example, '0' maps to the number 0, and '+' maps to the addition operation, etc.

15. (As I explained in my previous post) incompleteness implies that there are sentences true in the standard model but that are not members of the various theories. (Recall that 'member of a theory' just means 'provable in the theory'.) The only sentences that are unprovable from all consistent sets of axioms are sentences themselves that imply inconsistency.

If the theorem F1 is unprovable — TheMadFool

A theorem by definition is a provable sentence.

So what you wrote as the very first line of your argument is a contradiction in terminology.

And the rest of your argument is rife with other terminological mixups and misconceptions about mathematical logic.

It is irrational to mix up your own terminological use with the way the terminology is actually used for the mathematics you are critiquing.

If you wish to critique mathematical logic, then you should learn something about it first.

Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition).



"1. The theorem F1 is true AND the theorem F1 unprovable (Assume for reductio ad absurdum)"

By definition, a theorem is a provable sentence. So there is no F1 that is a theorem and unprovable.

So, to be generous, let's save 1:

F1 is true but unprovable.

"2. IF the theorem F1 is unprovable THEN we don't know the truth value of theorem F1"

False premise. A formula may be unprovable but known to be true by the method of models in the meta-theory.

"3 The theorem F1 is true (1 Simp)"

Okay.

"4. IF the theorem F1 is true THEN we know the truth value of theorem F1"

False premise. There are true formulas that we don't happen yet to know the truth value.

"5. We know the truth value of theorem F1 (3, 4 Modus Ponens)"

Incorrect since 4 is false.

"6. The theorem F1 is unprovable (1 Simp)"

When 1 is corrected as I have, then okay.

"7. We don't know the truth value of theorem F1 (2, 6 Modus ponens)"

Incorrect since 2 is false.

"8. We know the truth value of theorem F1 AND We don't know the truth value of thoerem F1 (5, 7 Conj)"

Incorrect since neither 5 nor 7 have been shown.

"9. False that the theorem F1 is true AND the theorem F1 is unprovable (1 to 8 reductio ad absurdum)"

Incorrect since 8 has not been shown.

I think there are vast more of those mathematicians who push the theorem to the sidelines close to the border of logic and logical inquiry and insist that it has nothing to do with anything else in the field of math than what the theorems state — ssu

Not quite. It has the respect of most in the math community, but most of those think they will never come up against that roadblock. Me included. It has arisen, however. For example Goodstein's Theorem.

Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition). — TonesInDeepFreeze

I'm not so sure but I don't think the problem has been solved. The Godel sentence, G = This theorem, T, is true but unprovable in the axiomatic system A. G makes the claim that T is true in A and not some other axiomatic system. For that, it's necessary for T to be provable in A.

You are completely confused about this subject.

What is the source you read about this subject?

In all these posts about incompleteness, by 'true' I mean 'true in the standard model'.

(1) Sentences are not true or false in a theory. Rather, sentences are true or false in a model for the language of the theory.

(2) G is true but G is not provable in the theory.

(3) G says that the Godel-number for G is not the Godel-number of a theorem of the theory. In other words, G says that G is not provable in the theory.

(4) G is true if and only if G is not provable in the theory.

(5) G does NOT say anything about its own truth value. Indeed, if a theory had a predicate for truth, then the theory would be inconsistent.

And don't overlook that your claimed reductio ad absurdum was refuted:

https://thephilosophyforum.com/discussion/comment/513487

What relationship does this topic have to philosophy, if any?

You are completely confused about this subject. — TonesInDeepFreeze

You maybe right but you didn't answer my question. Why? Please reread your reply to my question and my response to it.

Look, I'm not a mathematician and so you may find my comments rather strange but there seems to be an issue with Godel's theorems that your reply to my reductio ad absurdum argument highlights.

You said:

Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition). — TonesInDeepFreeze

Sentences are not true or false in a theory. — TonesInDeepFreeze

??? :chin:

I'll leave it to you to connect the dots.

I asked you:

What is the source you read about this subject? — TonesInDeepFreeze

You are using teminology and mentioning concepts in the subject, so it seems you've read something somewhere about it. Would you please tell me the articles or books you read? Then possibly I can look them up to find the passages you've misundertsood.

you didn't answer my question. Why? Please reread your reply to my question and my response to it. — TheMadFool

Why what? And if you have particular posts you want me to look back at then please link to them so I know specifically which ones you want me to look at.

The last qustion you asked is this:

the Godel sentence becomes, "proven" (as true) AND "unprovable". Isn't this a contradiction? — TheMadFool

No, it is not. I explained here:

https://thephilosophyforum.com/discussion/comment/513465

I have no clue whether you even read that post.

And I even followed up with extraordinary specifics here:

https://thephilosophyforum.com/discussion/comment/513476

But I'll say it again:

For convenience, let's suppose the theory is PA. And by 'true' I mean 'true in the standard model for the language of PA'.

G is not provable in PA. The sentence 'G is true' is provable in, e.g. Z set theory. That is, in Z set theory we state the standard model of PA and we can prove that G is true in that model.

Conflating provability with truth is one of the most common mistakes by people who only skim an article here and there about incompleteness and don't apprise themselves of the crucial technical specifics of mathematical logic and the incompleteness theorem.

I'll say it one more time:

G is provable in a theory

and

G is true in a model for the language of the theory

are DIFFERENT notions.

I'll leave it to you to connect the dots. — TheMadFool

If you have a question or a point to make, then please ask it or state it rather than dropping cryptic instructions for me to connect whatever dots I'm supposed to connect.

Perhaps you think there is an incompatibility here:

Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition).
— TonesInDeepFreeze

Sentences are not true or false in a theory.
— TonesInDeepFreeze — TheMadFool

I explained why those are not incompatible here:

https://thephilosophyforum.com/discussion/comment/513465

and with detailed definitions and points about mathematical logic here:

https://thephilosophyforum.com/discussion/comment/513476

And again in my post above.

And I'll say it yet another way:

Truth and provability are different notions in mathematical logic. While they are different, mathematical logic does study relationships between them. Some of those relationships are the subject of the incompleteness theorem.