Gödels Incompleteness Theorem's contra Wittgenstein

What Gödel proved was a truth that can only be seen in a formal system such as Peano Arithmetic.

The rules governing formal systems do not apply to informal systems,

where pretty much any instance of applying a new rule to an informal system, can be integrated into the system itself.[/quotes]

I don't know what that is supposed to mean.

[quote="Shawn;d15490"]There is no incompleteness in nature

Einstein had discussions with Gödel about how singularities and indeterminism could partake in physics and nature, which made him think nature was "incomplete," for lack of a better word).

until greater understanding is attained about the relation between logic and mathematics, which was an aspiration of many mathematicians and logicians during the war period, which Gödel had negated with his Incompleteness Theorems

if we can get past the conclusions of Gödel's Incompleteness Theorem's.

Gödel's incompleteness theorem applies to formal languages with countable alphabets.

So it does not rule out the possibility that one might be able to prove 'everything' in a formal system with an uncountable alphabet

OR expand the alphabet to account for new variables*.

No, Godel proved a meta-theorem regarding formal systems of a certain kind, including PA. The proof of that metatheorem can be done in various formal systems or done in ordinary informal mathematics, as is the case with Godel's original proof.

Moreover, the proof make use of only finitistic, intuitionistically acceptable principles.

Incompleteness is a property of certain formal systems. I don't know what it means to say that nature is or is not complete.

Einstein had discussions with Gödel about how singularities and indeterminism could partake in physics and nature, which made him think nature was "incomplete," for lack of a better word.
— Shawn

Who used the word, for lack of a better one? And what is your source?

What "conclusions" do you have in mind? The incompleteness theorem is a mathematical theorem with mathematical corollaries. Of course, some people make philosophical inferences based on the theorem, but such inferences are not of the mathematical theorem itself.

The defintion of 'formal language' includes that the language is countable.

There is no incompleteness in nature

This would imply that for every true statement about the physical universe, there exists a proof that can be derived from the supposedly canonical and categorical but unknown theory of the physical universe.

I do not like stating this in formal systems like Peano Arithmetic; but, rather in decidability. By framing the question in terms of decidability, we do away with the problem of the inherent limitations of human intuition devising formal systems. This is a question model theorists might be able to prove, in my opinion.

https://en.wikipedia.org/wiki/Decidability_(logic)

In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not.

The problem is logic itself:

I always thought the solution to the problem of insufficient logic needed to compute certain undecidable problems is solved by appealing to greater complexity class sizes, which avoids the inherent limitations of a formal system which is incapable of decidability given its inherent limitations.

An undecidable problem in logic is undecidable irrespective of how much time or memory you throw at the problem. The P versus NP issue only applies to problems that are at least logically decidable.

I think you are referencing Rosner.

it seems that it relies on a contradiction performed in the system

the liar paradox, which Rosner utilizes.

II hope that I might have gotten the gist of it.

Moreover, the proof make use of only finitistic, intuitionistically acceptable principles.
— TonesInDeepFreeze

I am not denying the logical validity of Godel's Incompleteness Theorems.

What Gödel proved was a truth that can only be seen in a formal system such as Peano Arithmetic.

the impact or the conclusions mathematicians reached at the time were too profound to the field of mathematics.

Well, at the time, many physicists were of the opinion that mathematics governs physics. So, I hypothesized that Einstein was aware of Godel's Incompleteness Theorems, since they had many discussions between each-other. I can only imagine that Einstein was interested in Godel's thoughts about physics according to Godel given his Incompleteness Theorems.

What "conclusions" do you have in mind? The incompleteness theorem is a mathematical theorem with mathematical corollaries. Of course, some people make philosophical inferences based on the theorem, but such inferences are not of the mathematical theorem itself.
— TonesInDeepFreeze

Sure; well, I won't comment on the effect Godel had on logic and mathematics. There are intellectuals that still ponder about it to this day, since this is a fundamental problem of mathematics at the time when Godel made his discovery.

The defintion of 'formal language' includes that the language is countable.
— TonesInDeepFreeze

There are potentially infinitely countable alphabets that could allow one to continuously expand the alphabet by including new terms in the formal system itself according to the principles of the formal system itself.

Maybe model theory can actually simulate reality with this possibility in mind.

If you allow for higher-order logic then all odds are off and even less can be asserted about the properties of the theories involved, such as incompleteness.

Sure; well, I won't comment on the effect Godel had on logic and mathematics. There are intellectuals that still ponder about it to this day, since this is a fundamental problem of mathematics at the time when Godel made his discovery.
— Shawn

You referred to conclusions that were drawn. But you don't have any in particular to mention.

I always thought the solution to the problem of insufficient logic needed to compute certain undecidable problems is solved by appealing to greater complexity class sizes, which avoids the inherent limitations of a formal system which is incapable of decidability given its inherent limitations.

I do not like stating this in formal systems like Peano Arithmetic; but, rather in decidability.

everyone concluded that this was the end of the possibility of proving everything in logic

It has not been demonstrated that propositional logic is the only logic that could accomplish the goals of unifying logic with mathematics or proving everything in logic alone.

Decidability for P verses NP is considered complete and consistent.

[the incompleteness theorem] was perceived by many as a hard limit on the ability to understand the world.

Not only has it not been demonstrated that propositional logic is not the only logic adequate for that task, but it's overwhelmingly clear that propositional logic is not adequate for that task.

What are complete and consistent, or incomplete or inconsistent are theories, not whatever "decidability for P v NP" is supposed to mean.

So one of the conclusions you are referring to is "incompleteness puts a hard limit on understanding the world"?

It is taken that incompleteness quashes Hilbert's program.

Are there any writers who you think are well paraphrased with that, and what writings of theirs do you have in mind?

one may be able to do so in come other formal language?

for a complexity class size to be complete and consistent, such as P v NP, then everything within such a set constitutes a complete and formal theory.

at the time Hilbert's program was one instance

given the assumption that a sufficiently sophisticated computable logical system with the capacity to compute with an ever expanding alphabet, in hypothetical terms, would be able to simulate reality. Again, this is an ad hoc argument against incompleteness

one could call propositional logic complete and even consistent for the complexity class size of P versus NP!

I don't know what that is supposed to mean, but, to be clear, the incompleteness theorem applies also to theories in higher order logic. Indeed, Godel's own proof regarded a theory in an omega-order logic.

https://en.wikipedia.org/wiki/Second-order_logic

This corollary is sometimes expressed by saying that second-order logic does not admit a complete proof theory. In this respect second-order logic with standard semantics differs from first-order logic; Quine (1970, pp. 90–91) pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not logic, properly speaking.

What are the definitions of 'complete for a complexity class size' and 'consistent for a complexity class size' such that a logic can be complete and consistent for a complexity class size?

I responded personally only when you falsely and snidely insinuated regarding my acquaintanceship with the halting problem and the method of definition.

And deserved though quite slight sarcasm about your claim about Einstein.

I have no comment on Wittgenstein.

I actually said that it was not meant to be a snide comment

To think that Einstein didn't have discussions about the import of the defining work of Godel, being his Incompleteness Theorems, would seem like a moot issue to profess skepticism over.

this thread was mostly about why Wittgenstein or what Wittgenstein could have meant by claiming that Godel's Incompleteness Theorems are logical tricks.

I will get decidedly personal at this point

You must mean that there's no point in you continuing.

Others can choose for themselves.

You must mean that there's no point in you continuing.
— TonesInDeepFreeze

No, you telling me at this point to simply 'shut up' won't happen, sorry.

no point in trying to cow down other members, which in this case is your personalization of the issue.

Amazing that you got that exactly backwards.

Like I said, I didn't get personal with you until you did with me.

Anyway, this thread was mostly about why Wittgenstein or what Wittgenstein could have meant by claiming that Godel's Incompleteness Theorems are logical tricks.

https://en.wikipedia.org/wiki/Remarks_on_the_Foundations_of_Mathematics

Particularly controversial in the Remarks was Wittgenstein's "notorious paragraph", which contained an unusual commentary on Gödel's incompleteness theorems. Multiple commentators read Wittgenstein as misunderstanding Gödel. In 2000 Juliet Floyd and Hilary Putnam suggested that the majority of commentary misunderstands Wittgenstein but their interpretation[8] has not been met with approval.[9][10]

Wittgenstein wrote:

I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable." Just as we can ask, " 'Provable' in what system?," so we must also ask, "'True' in what system?" "True in Russell's system" means, as was said, proved in Russell's system, and "false" in Russell's system means the opposite has been proved in Russell's system.—Now, what does your "suppose it is false" mean? In the Russell sense it means, "suppose the opposite is proved in Russell's system"; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by "this interpretation" I understand the translation into this English sentence.—If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in another game.)[11]

We usually know that a proposition is true because it is provable

If you look at what exactly Gödel's theorem says, There exist propositions in Russell's system that are (true and not provable) or (false and provable)

But if the system is sound, then the second disjunct is precluded.

It is possible to preclude the second disjunct if we assume or prove that PA is sound. I didn't say that PA itself proves that PA is sound. Virtually every mathematician (including Godel) regards PA to be sound.

What we do prove (in, for example, set theory) is that PA has model thus PA is consistent.

if PA is not sound, then it is actually unusable

However, proving soundness is even irrelevant.

Imagine that we prove soundness theorem.

No, because the proposition that proof implies truth is exactly what we are trying to prove.

If you look at what exactly Gödel's theorem says, There exist propositions in Russell's system that are (true and not provable) or (false and provable)

True but unprovable statements vastly outnumber the true and provable ones

The part that requires much proof is that the standard model is a model of PA.

"If a sentence P is provable from a set of sentences G, then all models of G are models of P"

If you look at what exactly Gödel's theorem says, There exist propositions in Russell's system that are (true and not provable) or (false and provable) — Tarskian


Where did Godel say that?

https://en.wikipedia.org/wiki/Diagonal_lemma

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions, and F(y) be a formula in T with one free variable. Then there exists a sentence C such that

T ? C ? F (?C?)

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Then there exists a sentence G such that

T ? G ? ¬ Bew(?G?)

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Then there exists a sentence G such that

G is (true and not provable) or G is (false and provable)

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

First Incompleteness Theorem: "Any consistent formal system T within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of T which can neither be proved nor disproved in T." (Raatikainen 2020)

There are denumerably many of each.

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.

(1) If PA is consistent, then there is a true but unprovable sentence.

there is no need to prove soundness theorem

Soundness can also be defined without using model theory:

https://en.wikipedia.org/wiki/Soundness

I am obviously not against using model-theoretical notions to define soundness

If you look at what exactly Gödel's theorem says

https://en.wikipedia.org/wiki/Diagonal_lemma

G is (true and not provable) or G is (false and provable)

This is somewhat equivalent to the alternative phrasing in which we assume consistency:

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

First Incompleteness Theorem: "Any consistent formal system T within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of T which can neither be proved nor disproved in T." (Raatikainen 2020)

not assume consistency

true and unprovable statements are non-denumerable.

In fact, by introducing the assumption "If PA is consistent", Gödel's theorem is no longer a theorem in PA. In that case, it is a theorem in PA + Cons(PA). That is not the same theory as PA.

That page relies on '|=' which is from model theory.

https://en.wikipedia.org/wiki/Logical_consequence

The turnstile symbol ? was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935).

If PA is consistent, then there are true but unprovable sentences. So, trivially, by disjunction introduction, it follows that there are true but unprovable sentences or there are false but provable sentences.

Meanwhile, for the third time, my remark is correct: If we assume soundness, then the second disjunct is precluded.

There are not only finitely many of them, and there are not uncountably many of them (there are only countably many sentences in the language), so there are denumerably many.

Incompleteness is not a theorem of PA, unless PA is inconsistent.

The use of logical entailment predates model theory by decades

T ?S as a synonym for "S is true in T".

I did not say that your remark would be wrong

it is not possible to preclude the second disjunct.

the Gödelian statements that cannot be expressed by language. There are uncountably many of those.

Gödel's incompleteness theorem proves that PA is inconsistent or incomplete.

That is a perfectly legitimate theorem in PA.

It does not prove that PA is incomplete.

That is a theorem in PA + Cons(PA).

hat is a perfectly legitimate theorem in PA.

I'm not sure about that; I'd have to think about it.

I don't think it's a theorem in PA, it's a theorem about PA.

I don't think it's a theorem in PA, it's a theorem about PA. PA + some additional axiom could make cons(PA) a theorem, but that wouldn't be a theorem in raw PA.

What would be more interesting is to understand why such implications arise within a formal system in the first place. Once we understand that, we can assess whether it’s reasonable to assume those implications might also hold for language or nature.

Did Wittgenstein even attempt to figure out why

PA |- ~Con(PA) v Inc(PA)

equivalently:

PA |- Con(PA) -> Inc(PA)

equivalently:

PA + Con(PA) |- Inc(PA)