Can a statement claim itself to be unprovable?

Question

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)

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Addendum

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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.

Answer 1

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.

Answer 2

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).

Answer 3

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.

Answer 4

"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.

Answer 5

"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.