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

[godelian]

I was always intrigued by the famous discussion in Russell's study concerning whether there was a rhinoceros (or something like that) in the room. Wittgenstein tried to convince Russell that he could not know for sure that there wasn't one Russell insisted that he could be sure, and it has been alleged that the two of them looked around the study for one just to prove their respective positions on the matter.

Technically, we cannot prove anything about the physical universe, if only because we do not have its theory -- to prove it from.

Furthermore, Russell was confusing possibility with probability.

No matter how improbable, there was nothing that allowed Russell to completely exclude the possibility.

Moreover, even if they found one in the room, there is no failsafe method to prove that to a third party. An image, a video, a witness deposition, they could all be fabrications.

What we know about the physical universe is at best probabilistically true. So, technically, I feel compelled to side with Wittgenstein.

Generally spoken, Russell was too sure of himself.

His grand masterwork, Principia, was bombastic and overly confident.

Russell lacked humility.

Russell was admired mostly by people who had never read what he had written because they did not understand it to begin with.

A great scientist does not believe in scientism -- unlike Russell.

Russell's naive activism was plain stupid.

[Harbal]

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?

What is the 'true statement', which is unprovable?

"This statement is not provable"

I'm guessing he came up with it to demonstrate that just because a sentence or statement conforms to the rules of grammar, that does not, in itself, make it meaningful.

[Gary Childress]

A great scientist does not believe in scientism -- unlike Russell.

I agree. Though I suppose the attempt has its merits, if only to have advanced human understanding by showing the falsity of the hypothesis.

[Gary Childress]

What is the 'true statement', which is unprovable?

"This statement is not provable"

I'm guessing he came up with it to demonstrate that just because a sentence or statement conforms to the rules of grammar, that does not, in itself, make it meaningful.

Strictly speaking, it seems to be a meaningful statement. It's says that it is "not provable". I suspect he mostly did it to prove that there are some statements which can be true which cannot be proven to be true. Though, I'm not sure how successful he was. It has a kind of puzzle like aspect to it. Philosophers seem to be known for their puzzles and paradoxes. I think of Zeno's paradoxes of movement and space as an example.

[Harbal]

"This statement is not provable"

I'm guessing he came up with it to demonstrate that just because a sentence or statement conforms to the rules of grammar, that does not, in itself, make it meaningful.

Strictly speaking, it seems to be a meaningful statement. It's says that it is "not provable".

What is in the statement that is subject to proof? There is nothing there, as far as I can see; it's just a meaningless sentence, and I think that is the point. You assume the sentence must mean something because it makes grammatical sense, so you tie yourself in knots looking for logical sense that simply isn't there. That's my opinion, anyway, and I think it teaches a valuable lesson.

[Gary Childress]

I'm guessing he came up with it to demonstrate that just because a sentence or statement conforms to the rules of grammar, that does not, in itself, make it meaningful.

Strictly speaking, it seems to be a meaningful statement. It's says that it is "not provable".

What is in the statement that is subject to proof?

That's a good question. I suppose ""this statement is not provable" is not provable." is what is meant. Then we get into an infinite regress perhaps of whether or not one can prove whether or not "this statement is not provable, is not provable, is not provable" is provable or not. It's a complete statement strictly speaking. But it's not a statement like "all bachelors are unmarried men". So you may have a point there.

[Harbal]

Strictly speaking, it seems to be a meaningful statement. It's says that it is "not provable".

What is in the statement that is subject to proof?

That's a good question. I suppose ""this statement is not provable" is not provable." is what is meant. Then we get into an infinite regress perhaps of whether or not one can prove whether or not "this statement is not provable, is not provable, is not provable" is provable or not. It's a complete statement strictly speaking. But it's not a statement like "all bachelors are unmarried men". So you may have a point there.

Well I do have some experience with meaningless sentences, Gary.

[Gary Childress]

What is in the statement that is subject to proof?

That's a good question. I suppose ""this statement is not provable" is not provable." is what is meant. Then we get into an infinite regress perhaps of whether or not one can prove whether or not "this statement is not provable, is not provable, is not provable" is provable or not. It's a complete statement strictly speaking. But it's not a statement like "all bachelors are unmarried men". So you may have a point there.

Well I do have some experience with meaningless sentences, Gary.

I think you are very intelligent, Harbal. I also think you are a dynamic thinker, not one who is locked into a particular tunnel of vision as it were.

[Harbal]

That's a good question. I suppose ""this statement is not provable" is not provable." is what is meant. Then we get into an infinite regress perhaps of whether or not one can prove whether or not "this statement is not provable, is not provable, is not provable" is provable or not. It's a complete statement strictly speaking. But it's not a statement like "all bachelors are unmarried men". So you may have a point there.

Well I do have some experience with meaningless sentences, Gary.

I think you are very intelligent, Harbal. I also think you are a dynamic thinker, not one who is locked into a particular tunnel of vision as it were.

That means quite a lot coming from you, Gary, thanks.

[godelian]

That's a good question. I suppose ""this statement is not provable" is not provable." is what is meant. Then we get into an infinite regress perhaps of whether or not one can prove whether or not "this statement is not provable, is not provable, is not provable" is provable or not. It's a complete statement strictly speaking. But it's not a statement like "all bachelors are unmarried men". So you may have a point there.

There are three possibilities: the statement "this statement is not provable" (K) is true, false, or undecidable.

- If K is true, then it is a true but unprovable statement.
- If K is false, then it is a false but provable statement.
- K is undecidable.

The incompleteness theorem says: There exist true unprovable or false provable or undecidable statements.

Conclusion: K is indeed an example case ("witness") for the theorem.

By the way, it is by no means considered to be a "good" example. In the meanwhile, much better examples have been discovered.

[Harbal]

Trawled from the internet:

The statement "This statement is not provable" is an example of a self-referential or self-referential paradoxical statement. It belongs to a class of statements that refer to themselves in a way that creates a logical contradiction or paradox.

In this case, if the statement is provable, then it must be false because it claims to be unprovable. However, if it's unprovable, then it must be true, which contradicts the claim that it's unprovable. This creates a logical loop or paradox.

The statement is closely related to Gödel's incompleteness theorems, which are two theorems of mathematical logic that establish inherent limitations in formal axiomatic systems. Gödel's second incompleteness theorem states that no consistent formal system can prove its own consistency, which can be loosely related to the statement "This statement is not provable."

[godelian]

Trawled from the internet:

However, if it's unprovable, then it must be true, which contradicts the claim that it's unprovable.

The fact that it is true does not contradict the claim that it is unprovable.

Why would a true statement necessarily be provable?

The fact that a statement is true will by no means give you access to a proof. It is not a logical paradox. He merely confuses truth with provability.

[godelian]

I'm guessing he came up with it to demonstrate that just because a sentence or statement conforms to the rules of grammar, that does not, in itself, make it meaningful.

Say that any logic statement could be true, false, or undecidable.

So, "This is unprovable" is true, false, or undecidable.

So, "This is unprovable" is (true and unprovable), (false and provable), or undecidable.

Look carefully at what Godel's incompleteness theorem claims:

There exist logic sentences that are (true and unprovable) or (false and provable) or undecidable.

Isn't it therefore obvious that "This is unprovable" is a legitimate witness for the theorem?

There are more and arguably better witnesses, but that is another discussion altogether.

[Gary Childress]

I'm guessing he came up with it to demonstrate that just because a sentence or statement conforms to the rules of grammar, that does not, in itself, make it meaningful.

Say that any logic statement could be true, false, or undecidable.

So, "This is unprovable" is true, false, or undecidable.

So, "This is unprovable" is (true and unprovable), (false and provable), or undecidable.

Look carefully at what Godel's incompleteness theorem claims:

There exist logic sentences that are (true and unprovable) or (false and provable) or undecidable.

Isn't it therefore obvious that "This is unprovable" is a legitimate witness for the theorem?

There are more and arguably better witnesses, but that is another discussion altogether.

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?

[godelian]

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.