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On Wiki, the Second Incompleteness Theorem reads as

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.

I don't know mathematical logic, and probably I don't get the actual message of the SIT, but is it possible to prove the consistency of $T$ in another theory $T_1$, and recursively find $T_{n+1}$ proving $T_n$ consistent, until $T_N$ doesn't fulfil the hypothesis of the SIT and can prove its own consistency?

Andrés E. Caicedo
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Bananach
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    No, it is not possible in any useful way. If $T_1$ satisfies the assumptions of the theorem, and $T\supset T_1$, then, as long as $T$ is effectively generated and consistent, $T$ does not prove its own consistency either. The only way to get around this is to have $T$ either inconsistent (and therefore useless) or not effectively generated. In the latter case, we may not even be able to express that $T$ is consistent as a formal statement. And if we manage to circumvent this issue, given a statement $P$, in general we cannot even determine whether $P$ is one of the axioms of $T$ or not. – Andrés E. Caicedo Sep 06 '13 at 01:45
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    There are other obstructions, but more technical. For example, as long as we have an effective procedure to pass from $T_n$ to $T_{n+1}$, then the whole $\bigcup_n T_n$ also satisfies the assumptions of the result. – Andrés E. Caicedo Sep 06 '13 at 01:47
  • Thanks! Especially your second comment opened my eyes, I think :) – Bananach Sep 06 '13 at 15:04

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