Why did Godel come up with a true statement that is unprovable?

[commonsense]

Was it just because he could? Or was there some kind of need for it? And if it was because there was a need for it, what was that need?

Was it not to support his Incompleteness Theorem? I.e. in any closed system there are statements that cannot be proved by other statements from the same system. Perhaps many of the posts here on the unprovability of a statement further underscore the wisdom of his theorem.

[Gary Childress]

What are the ramifications of the theorem? What does this mean for logic, or what's the significance of there being some statements that are true but not provable?

First of all, the second half of the theorem is routinely ignored: Or there are statements that are false and provable.

This is a very damaging eventuality because it suggests that logic is potentially inconsistent.

According to the second incompleteness theorem, if a system can prove that it is consistent, then it is necessarily inconsistent.

We are truly in the middle of an exercise in disaster tourism. It is a tour of Chernobyl nuclear reactor 4.

There is no guarantee that logic is consistent. We routinely assume that it is, but we do so without any assurance that it is true.

The "true but not provable" part points out that arbitrary, inexplicable truths exist even in perfectly deterministic systems. A system does not need any randomness whatsoever to be largely unpredictable.

We don't know if randomness is truly random -- or uberhaupt even exists -- because it is not a requirement for unpredictability. In fact, we already suspected this because pseudorandom number generators routinely pass all tests for randomness while not being random at all.

The universe could very well be both deterministic and unpredictable.

When you say deterministic and unpredictable is it possible that the universe could generate "randomness" in such a way as that over an infinite amount of time, the same deterministic sequence would begin to repeat the exact sequence over again? In other words, randomness may possibly only be the appearance to humans who are unable to sufficiently count high enough to know what the determined sequence ultimately is? Because I believe computers (at least back in the day, maybe this has been improved) used to generate "random" numbers but they were based on incredibly large deterministic lists of numbers being selected in sequence over and over again, but that were extremely difficult for a human to determine where in the sequence they were and therefore the observer was unable to predict what the next number would be.

[godelian]

When you say deterministic and unpredictable is it possible that the universe could generate "randomness" in such a way as that over an infinite amount of time, the same deterministic sequence would begin to repeat the exact sequence over again?

Yes, if you repeat the history of the universe from its beginning, it is possible that you see the same history again.

A good number of seemingly unpredictable observations may not be random. They cannot be predicted from the theory of the universe but they may not be freely occurring either.

This is the case for facts in the universe of the natural numbers. Unpredictable facts exist, even though randomness does not exist in the universe of the natural numbers. The physical universe could have similar properties.

[learis]

"This statement is not provable"

Definition of a Statement:
From Internet Encyclopedia of Philosophy:
A statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false.


A statement by definition must be capable of having a truth value. To state that this statement is not provable (aka not capable of having a truth value) is to go against the very definition of what a statement is.

So the original quote is absurd & malformed. There's nothing to debate.

[godelian]

"This statement is not provable"

Definition of a Statement:
From Internet Encyclopedia of Philosophy:
A statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false.


A statement by definition must be capable of having a truth value. To state that this statement is not provable (aka not capable of having a truth value) is to go against the very definition of what a statement is.

So the original quote is absurd & malformed. There's nothing to debate.

"Provable" means that there exists a proof somewhere in the database of proofs for the statement at hand.

Imagine that you remove its proof from the database, does that mean that the statement suddenly becomes false? No, of course not! It remains true. It is just not provable anymore.

In 1931, Godel did of course not use the term "database", but he still went to great length to construct an abstract one, along with a lookup function. He used it to abstractly implement the predicate isProvable(s).

So, you try to look up the proof in the database. If it exists, then the statement is provable. Otherwise, it is not.

Hence, provability is clearly distinct from truth.

The confusion actually stems from Soundness theorem. We routinely assume Soundness theorem:

If a statement is provable from its theoretical context, then it is true in all its interpretations ("models").

For first-order logic, Soundness theorem is also confusingly known as Godel's (semantic) completeness theorem.

The reverse is not necessarily the case.

It is not because a statement is true (in a particular interpretation) that it is provable from its theoretical context. This is bound to happen if the statement is true in one interpretation but not in another one. This is known as incompleteness.

In first-order arithmetic, because of Lowenheim-Skolem theorem, we know that there are an infinite number of nonstandard models ("interpretations") besides the standard one, i.e. the natural numbers.

Therefore, the suspicion existed already that in first-order arithmetic there were statements true in one model but not in another (nonstandard) one. Godel finally proved this.

If you confuse truth with provability, you cannot possibly understand Godel's work.