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Given some theorem $T$, could the question "is $T$ decidable?" be undecidable?

I assume the answer is yes, and if it is, could the decidability of a theorem be undecidable even if the theorem itself is decidable?

Ty Jensen
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  • If a theorem is decidable, i.e. provable or disprovable, then that proof/disproof will be a proof of decidability. So decidability can be undecidable only for undecidable theorems. – Conifold Aug 30 '21 at 06:20

1 Answers1

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If T is undecideable within formal system F, it means (1):

  • F does not contain a proof of T, AND
  • F does not contain a proof of ~T.

Suppose the question "T is undecideable" is undecideable. That means (2):

  • F does not contain a proof that T is undecideable in F, AND
  • F does not contain a proof that T is not undecideable in F.

In particular, note the second dot point means (3):

  • F does not contain a proof that F contains a proof of T, AND
  • F does not contain a proof that F contains a proof of ~T.

Let's also assume that T is not undecideable. That means (4):

  • F contains a proof of T, OR
  • F contains a proof of ~T.

Either way, this contradicts (3), since if F contains a proof P of T, we can construct a proof that F contains a proof of T: that would be to note P, and check that it is a proof (which can be done if F is sufficiently powerful for godelian arguments to work). Likewise, if F contains a proof of ~T.

Michael Hartley
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  • Proofs that T is undecidable are not given in F itself, they are given in a meta-theory. F typically does not even have expressive means to talk about its own theorems. – Conifold Aug 30 '21 at 06:13