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Given that according to Gödel's theorems there are propositions in any language equivalent to first order logic that cannot be proven right or wrong, is it possible to prove that some of such propositions are not provable? I am thinking of something like a proof of impossibility applied to propositions.

Francesco Bertolaccini
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  • Gödel's incompleteness theorems are in some sense impossibility theorems. But you apparently don't mean those. What do you mean? – Mees de Vries May 02 '18 at 13:26
  • Gödel states that there are propositions that cannot be proved. But can you prove that a specific proposition can not be proved? – Francesco Bertolaccini May 02 '18 at 13:27
  • Yes, that is Gödel's second incompleteness theorem; it states that an (appropriate for for the theorem) theory cannot prove its own consistency. E.g., $\mathsf{PA} \not \vdash \mathrm{Con}(\mathsf{PA})$. – Mees de Vries May 02 '18 at 13:30
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    Gödel states that for a specific formal system $F$ there is a proposition $G_F$ (a formula in the language of $F$) that cannot be proved in $F$. $G_F$ is "specific" in the sense that we can "build" it. Of course, it makes little sense to ask about "absolute" impossibility: we can simply add $G_F$ to the system $F$ as axiom to get a new system $F'$ and $G_F$ is provable in $F'$. – Mauro ALLEGRANZA May 02 '18 at 13:32
  • Interesting, those could be answers. But is there any example of the theorem being applied to other proofs? Could it be used to prove that some conjectures are unsolvable? – Francesco Bertolaccini May 02 '18 at 13:34
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    This might help https://math.stackexchange.com/q/2027182/471959 – ℋolo May 02 '18 at 13:35
  • @holo so the general strategy is to find a statement which contradicts the starting one, and prove that both are compatible with your existing set of axioms? – Francesco Bertolaccini May 02 '18 at 13:39
  • It opened the way to proving that some "interesting" results cannot be proved in some specific system but needed "stronger" ones; see e.g. Goodstein's theorem. – Mauro ALLEGRANZA May 02 '18 at 13:40
  • @Mauro thanks, that was one of the examples I was looking for! – Francesco Bertolaccini May 02 '18 at 13:42
  • You are welcome :-) – Mauro ALLEGRANZA May 02 '18 at 13:44
  • @FrancescoBertolaccini something like this, but doing so it is not something that can easily done. – ℋolo May 02 '18 at 13:44

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Gödel showed how to construct a statement in any sufficiently powerful formal system that states its own unprovability within the language of the formal system itself (he did this by showing that propositions in the language of the formal system can be translated into numbers and that "provability" can be translated into artithmetical properties of those numbers). But his construction method creates a specific proposition - it cannot be used to determine the provability of a general proposition.

This is analagous to how Liouville showed that transcendental numbers exist by constructing a family of numbers with properties which allowed him to prove they were not algebraic. It is much more difficult to prove that a particular number such as $e$ or $\pi$ is transcendental.

gandalf61
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Impossibility of a proof in generality means: the statement is false.

In mathematics all true statements are provable.

Considering mathematics as an abstract model of certain aspects of reality, a statement being true means: it's consistent with all other statements of the model.

If a statement is consistent, there are ways to show it, to construct a proof.

As for Gödel, something not provable "within the language of the formal system itself", that will happen, as no sentence may prove themselves.

Translating Gödels argument into building a house, into construction, may read:

When building a house, the correctness of its construction might may not be stated completely from within.

Correct. But, why not go outside and check it from some distance?

Andreas Röhler
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