If so then there is at least one x such that x = "x is unprovable":

1) x = "x is unprovable"

It must now be true that:

2) x is provable IFF "x is unprovable" is provable

Since only true sentences are provable simplification gets a contradiction:

3) x is provable IFF x is unprovable

And we must conclude that there is no statement claiming itself to be unprovable. (QED)


As to formal systems: They may perhaps produce statements saying that they are unprovable in the formal system ... which is true if the system is consistent.

That is a restricted concept of unprovability.

It has been said that the term "unprovable" is vague.

That it is relative and needs a qualification such as "Unprovable by ruler and compass" ...
But I think there is a basic meaning that can be defined:

(definition) x is unprovable IFF there can be no proof of x.

  • Gödel's theorem exactly is such a statement. Do you try to prove it's wrong? – rus9384 May 19 '18 at 14:28
  • Gödels formula is not an unprovable formula: It does NOT claim itself to be unprovable! It claims that it is not provable in Peano Arithmetics ... Which we can see is true or Peano Arithmetics is inconsistent. – Sigurd Vojnov May 19 '18 at 14:42
  • Claiming something != demonstration nor proof of claim. – Mr. Kennedy May 19 '18 at 14:52
  • No, Gödel's Theorem can't be formulated in too weak arithmetics, where all true statements are provable. And any arithmetics where Gödel's Theorem can be formulated, has unprovable statements. – rus9384 May 19 '18 at 14:53
  • 4
    provable is relative to a context (a theory). Speaking of "absolute" provability is vague. – Mauro ALLEGRANZA May 19 '18 at 15:08

The mistake here is that 2) and 3) are not equivalent. Not all true statements are provable, so you can't simplify "x is provable" to "x is true", since it is a stronger statement to say y IFF x provable than to say y IFF x.

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Godel's incompleteness theorem essentially states that any formal system complicated enough to create a statement like "This statement is not provable" is either inconsistent (some statements are both true and false) or incomplete (some true statements are unprovable within the system).

It seems that you are trying to get around this problem by redefining what is allowed to be a statement. One could certainly alter a logical system by adding the rule that any valid statement must be provably true or provably false. Such a system would no longer be incomplete, though it might become either inconsistent or trivial (no statements exist).

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  • 1
    "this statement is unprovable" cant say that it itself is unprovable because then it must be the case that 1) this statement is identical with "this statement is not provable". But if so then 2) this statement is provable IFF "this statement is not provable" is provable. Provable sentences are true so we get. 4) this statement is provable IFF this statement is not provable The result is that "this statement is unprovable" cant be used to say that it itself is unprovable. Your formal system can only produce sentences saying that the formal system cant prove them. – Sigurd Vojnov May 24 '18 at 18:28

2) x is provable IFF "x is unprovable" is provable

Following the defintion in Premise 1, Premise 2 is saying the following: the hypothesis that Statement X is unprovable is itself provable if and only if it is possible to prove that such hypotheses are provable. The statement is actually closer to a tautology.

Premise 2 does not assist the conclusion that Statement X is provable if and only if Statement X is unprovable.

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"P is unprovable" is a poor attempt at the Liar Paradox. "P is unprovable" if it is logically unprovable (for instance, P is a metaphysical proposition) or by enumeration, it is justifiable to assert "P is unprovable" (such is the case with "I exist").

"This statement is false" is the introductory statement to how language and propositions work, and most importantly, that the LNC is a strategy rather than a law (as Dialetheism attests to).

Given the following, "This statement is false" doesn't express a proposition:

P = [content][is or is not the case]

b = [it is true that] P

d = [it is not true that] P

a = b or d [as declarative speech act]

Where P is "proposition", b is "belief", d is "doubt", and a is "assertion".

The dialectic analysis of P (according to Peirce, Russell, and others) is:

P = [This sentence][is false]

And so, "This sentence" doesn't purport to be about any state of affairs and then cannot be true or false.

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"Proof" is a vague word. I'd say that a proof is an ordered set of transformations of statements starting with premises and ending with the conclusion. It is therefore necessary to have specific premises and legal transformations, which pretty much means that proofs have to be in formal systems.

It is possible to have other definitions of proof, I suppose: "I tell you once, I tell you twice/What I tell you three times is true" could be taken as a method of proving things (although not a really useful one). In that case, anything that can be pronounced can be proven, including 'This statement is unprovable." when you repeat it three times.

So, what do you mean by "proof"? It has a clear meaning in a formal system, and in a formal system it is possible to come up with "Statement X: X is unprovable." If you consider proof outside a formal system, you still need to put some limits on it.

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