Gödels Incompleteness Theorem's contra Wittgenstein

Shawn

Another one of my topics in the Logic & Philosophy of Mathematics section.

Wittgenstein as a logician stated that the sum total of logic consists of tautologies, contradictions included. He also said that quintessentially because logic is tautological, logic takes care of itself.

Now, both Wittgenstein and Gödel met at the Vienna Circle, a meeting-place for the then acclaimed logical positivists. After Wittgenstein published his Tractatus-Logico-Philosophicus, the Vienna Circle studied it not only once but twice. Gödel held a positive opinion of the work. The only point of contention between Gödel and Wittgenstein was one single instance of disagreement about the implication of Gödel's work on logic and mathematics. Namely, Gödel's Incompleteness Theorems is what Wittgenstein claimed to be logical "tricks". Some people say that Wittgenstein didn't understand Gödel, however, I think it is the other way around. Let me explain in brief.

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. This feature of informal systems, such as language, is not possible in formal systems like Peano Arithmetic. Wittgenstein went to great lengths in the Philosophical Investigations to try and see what rules govern language, and was not able to provide a clear answer. Just to point out my own understanding, the problem of what governs rules in language is a subject best framed, not in terms of logic, but according to syntax and grammar. Grammar is not subject to the constrains of logic, it takes part in the very structure of how language is able to be transmitted and understood between people. One day we will be able to understand universal grammar, which would be able to answer the question that grappled or still grapples many analytic philosophers since Bertrand Russell and his protege, Wittgenstein.

To return to the topic of Wittgenstein and Gödel, if one looks even at nature or physics, which describes logic, which the system of nature and physics is in fact complete and whole. There is no incompleteness in nature (As a side note, 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). This ad hoc argument is formulated against the Incompleteness Theorems regarding how nature and the universe are unitary and whole. None of this would make any sense to readers who think that logic is independent of reality. I don't think this is possible. Logic manifests itself from human intuition and the laws of nature describing the relations logic can posit, through human understanding.

Not to sound grandiose; but 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, then we will not have the scaffolding to venture past the conclusions of Gödel. Hilbert's program was intended for such greater understanding. Again, logic takes care of itself and its relation to mathematics is quite possibly something we would be able to better understand if we can get past the conclusions of Gödel's Incompleteness Theorem's.

I would like to end this post with a thought that generated, mostly, these corresponding thoughts about Gödel and Wittgenstein, and physics.

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*. — Shawn

*-Variables could stand for new understanding itself, within a system of thought or a model of reality.

Edit: Added and fixed some things.

TonesInDeepFreeze

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

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.

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

Godel's proof pertains only to formal 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.
— Shawn

There is no incompleteness in nature — Shawn

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?

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 — Shawn

The incompleteness theorem is an important limitation of certain kinds of formal systems. But the incompleteness theorem doesn't "negate" the importance of logic in mathematics. The incompleteness theorem doesn't "negate" that virtually all (or all) of classical mathematics is axiomatized in a system that uses first order logic.

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

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.

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

A language is not formal if it is not a countable language. The incompleteness theorem pertains only to systems with formal languages.

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

There are no formal languages with uncountably many symbols. There are languages written in symbolic logic that have uncountably many symbols, but they are not formal languages. The defintion of 'formal language' includes that the language is countable.

OR expand the alphabet to account for new variables*. — Shawn

As long as you add only countably many variables, the incompleteness theorem will hold.

Shawn

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. — TonesInDeepFreeze

Yes, well I think you are referencing Rosner? Can you provide the reference?

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

Yes, I would like to point out that I am not denying the logical validity of Godel's Incompleteness Theorems. I am only suggesting that the impact or the conclusions mathematicians reached at the time were too profound to the field of mathematics.

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

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? — TonesInDeepFreeze

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 would be able to simulate reality with this possibility in mind. Regarding which:

4.01 [...] A proposition is a model of reality as we imagine it.

Tarskian

There is no incompleteness in nature — Shawn

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.

We do not know the theory of the physical universe.

We also do not know if it happens to be canonical or categorical.

In the context of the natural numbers, we know that Peano Arithmetic theory is not canonical (there are definitely alternatives) and not categorical either (it does not have a single interpretation/model).

All of this in the context of first-order logic. 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.

Shawn

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. — Tarskian

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

I'm interested in your take according to the halting problem being avoided 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, if possible. I'd also like to mention that this is only true for completeness, not consistency given that there actually seem to be singularities in the fabric of spacetime, according to Einstein, Hawking, and Penrose.

Tarskian

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. — Shawn

The problem is logic itself:

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.

Shawn

The problem is logic itself: — Tarskian

I always thought the solution to the problem of certain logical systems needed to compute undecidable problems, is solved by appealing to greater complexity class sizes. This allows for other more sophisticated/robust systems of logic to be discovered instead of the inherent limitations of a formal system which is incapable of decidability.

Tarskian

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. — Shawn

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.

Shawn

I'd like to add to my previous post, that the implications of incompleteness have been only demonstrated by incorporating new procedural rules into a system, which could not prove their use as consistent with regards to the principles of the formal system itself prior to adding the procedure. With that said, there could be formally complete systems that are consistent, which occupy their own logical space.

Now, everyone concluded that this was the end of the possibility of proving everything in logic, and even unifying logic with mathematics. I don't think this conclusion is the right one to draw. It would almost seem like the ad hoc ergo propter hoc fallacy.

One could always try and add new terms defining the formal system to add sophistication to the formal system itself to account for the new terms. 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.

Shawn

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. — Tarskian

Yes; but, returning to what was said in the OP, then I believe that if logic is tautological, then the only constraints on systems of logic are the ways in which we try and change the principles of the formal system itself. So, I don't think we really know that the field of logic is over and there is nothing more to discover about it. We could always discover a way in which logic behaves in terms of differing terms defining how different formal systems interact. Given that physics and reality is composed of dimensions, I don't think it is farfetched to say that logic is not a field that can be described through one system like propositional logic, and as you said one could call propositional logic complete and even consistent for the complexity class size of P versus NP!

TonesInDeepFreeze

I think you are referencing Rosner. — Shawn

Maybe you mean Rosser. Rosser improved Godel's theorem, but that has nothing to do with what I said in my post.

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

That is very wrong and backwards. No contradiction is shown in the system. Rather the system is assumed to be consistent*. From the assumption that the system is consistent*, formal and sufficient for arithmetic, we derive that the system is incomplete.

*Here, Rosser does play a role. For Godel it's w-consistent but for Rosser it's plain consistent.

the liar paradox, which Rosner utilizes. — Shawn

The incompleteness theorem does not use the liar paradox. The incompleteness uses something only analogous to the liar paradox. The liar paradox is "this sentence is false". Incompleteness uses "this sentence is unprovable". "This sentence is false" leads to contradiction. "This sentence is unprovable" does not lead to contradiction.

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

You didn't. You got it very wrong.

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

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

The theorems are not validities. Rather, the proofs are valid. The theorems are not true in every model. Rather, if the proof is formal, then there are no models in which the axioms used for the proof are true and which the theorem is not true.

And what I responded to in this context is this:

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

That is wrong.

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

What specific mathematicians and conclusions are you referring to? And how the conclusions too profound?

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

"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?"— TonesInDeepFreeze

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. — Shawn

So you don't know that Einstein said that nature is incomplete. You only imagine that he did. You imagine it on the basis that Einstein discussed the incompleteness theorem with Godel. That's compelling.

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. — Shawn

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

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. — Shawn

For every countable language L, there is another countable language K that is a proper superset of L. So what?

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

Whatever that is supposed to mean, adding countably many symbols to a countable language doesn't bear upon the incompleteness theorem. And adding countably many variables has no substantive effect on a language.

TonesInDeepFreeze

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. — Tarskian

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.

Shawn

Not to disregard what you said; but, thank you for the clarification. I edited that post which you addressed because it was inaccurate.

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. — TonesInDeepFreeze

There are so many things various people have said about Godel's Incompleteness Theorems, which is telling about how mostly logicians and mathematicians have been influenced by Godel. What Godel said was so fundamental to the field of mathematics and how it was perceived by many as a hard limit on the ability to understand the world. I'm not a mathematician or logician, and am quite a novice at logic; but, I think that the implications of Godel on the future of computing hasn't really been addressed. My opinion about this is that as we advance our understanding of complexity class sizes in computing, then it seems possible that new ways of inference could be discovered. Logic has been a field most appreciated by computer scientists, along with the father of computing, Turing, describing Godel's Incompleteness Theorems as the halting problem.

TonesInDeepFreeze

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. — Shawn

I don't know what that is supposed to mean. But incompleteness is not avoided by greater and greater proof capability. No matter how capable, if the system is formal, consistent and provides for arithmetic, then it is incomplete.

TonesInDeepFreeze

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

What do you mean by 'decidable'? 'decidable' in context of incompleteness and computability theory has a definition. Is yours the same?

Shawn

↪TonesInDeepFreeze

Decidability for P verses NP, in computer science, is considered complete and consistent.

TonesInDeepFreeze

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

It is taken that incompleteness quashes Hilbert's program. I think it might depend on one's definition of 'logicism' whether incompleteness also quashes logicism.

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. — Shawn

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.

TonesInDeepFreeze

Decidability for P verses NP is considered complete and consistent. — Shawn

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

TonesInDeepFreeze

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

So one of the conclusions you are referring to is "incompleteness puts a hard limit on understanding the world"? Are there any writers who you think are well paraphrased with that, and what writings of theirs do you have in mind?

Shawn

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. — TonesInDeepFreeze

So, one may be able to do so in some other formal language?

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

I'm of the understanding that 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.

Shawn

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

It is taken that incompleteness quashes Hilbert's program. — TonesInDeepFreeze

Well at the time Hilbert's program was one instance, as you say.

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

I don't have any to provide.

TonesInDeepFreeze

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

Depends on what you mean by "everything".

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. — Shawn

We were talking about incompleteness, in which context 'complete' and 'consistent' have certain definitions. So I couldn't follow what might be your intent in using them in another sense. I don't know what definitions of them you have in mind related to complexity. It seems there are such definitions, but I don't know enough about them.

TonesInDeepFreeze

at the time Hilbert's program was one instance — Shawn

What's another? If it's just Hilbert's program, then why not just say that from the start? But then your reasoning about that limitation in connection with incompleteness was ill-premised and confused about what the incompleteness theorem even is.

TonesInDeepFreeze

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 — Shawn

How so? How would such a thing show that it is not the case that every consistent, formal theory sufficient for arithmetic is incomplete?

TonesInDeepFreeze

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

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?

Tarskian

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. — TonesInDeepFreeze

What I meant to refer to, was:

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.

TonesInDeepFreeze

↪Tarskian

Two different senses of 'complete':

(1) a theory T is complete iff for every sentence P in the language of T, either P is a theorem of T or ~P is a theorem of T

(2) a logic L is complete iff for every valid formula P, P is a theorem of L

The incompleteness theorem concerns (1).

The completeness theorem is that first order logic is complete in the sense of (2).

Higher order logic, if formalized, is not complete in the sense of (2); that is, there is no formal set of axioms that proves all the validities.

EricH

↪TonesInDeepFreeze

I’m continually impressed with how patient you are in these discussions.

TonesInDeepFreeze

↪EricH

It seems that most often I'm not patient. In these situations, patience is a fault.

Shawn

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? — TonesInDeepFreeze

I don't mean to be snide; but, if you haven't acquainted yourself with the halting problem, then it would clarify any confusion about how definitions are stated and applied.

TonesInDeepFreeze

↪Shawn

You don't know what you're talking about:

(1) Of course I know the halting problem. You make no point with the false insinuation that I don't.

(2) I know a lot better than you how definitions are stated and applied, since I've actually studied the details about definitions in logic, especially the forms that provide for eliminability and non-creativity.

(3) I don't claim that there are not 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, nor that, even if your verbiage is not exact then it still couldn't be explicated. I merely asked for your definitions. If you don't have definitions to share or have them but don't wish to share them, then so be it. But without them, you cannot reasonably expect another to appreciate your arguments. And shifting to a phony insinuation about my acquaintance with the halting problem caps you as a fool.

(4) My best guess is that you're throwing around a lot of technical verbiage for effect. And the basis for that guess is that, from your first post, you've been flat out incorrect about the incompleteness theorem.

(5) I do mean to be snide; if you haven't acquainted yourself with what the incompleteness theorem actually is, then it's likely that your confusions and false statements about it will persist.

Gödels Incompleteness Theorem's contra Wittgenstein

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