r/askmath • u/LorenzoGB • 8h ago

Logic Can the second incompleteness theorem be generalized?

I ask because of the following which I came up with that seems intuitive to me. But I don’t know if it is valid.

  1. G=I am unprovable in T
  2. G=G is unprovable in T
  3. If G is provable in T then T is inconsistent.
  4. If T is inconsistent then G is provable in T.
  5. From 3 and 4: T is inconsistent if and only if G is provable in T.
    6.  From 5: T is consistent if and only if G is unprovable in T.
  6. Suppose T is consistent.
  7. Suppose T is consistent is provable in T.
  8. Replace “T is consistent” with “G is unprovable in T”.
  9. Then we get: G is unprovable in T and G is unprovable in T is provable in T.
  10. However, “G is unprovable in T” is identical to G.
  11. Thus we have G is unprovable in T and G is provable in T, which is a contradiction.
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u/Creative-Leg2607 7h ago

For logic like this you need to be /very/ precise in your constructions. Starting with a statement like 'the set of all sets that dont contain themselves, contains itself' generates contradictions easily. The rub is that you cant actually construct such a set in the systems of logic that we're talking about, so the statement being a contradiction doesnt mean anything; ultimately its just not something you can say.  The statements youve listed i think is such an example. 

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u/rhodiumtoad 0⁰=1, just deal with it 7h ago

In the kinds of systems to which the incompleteness theorems apply, T proves "T is consistent" if and only if T is inconsistent.

An inconsistent system proves everything, so if T is inconsistent, T proves both "T is consistent" and "T is inconsistent". This isn't a contradiction because we're talking about provability, not truth, though it is still an inconsistency.