On some generalizations of G\"{o}del's second incompleteness theorem

In this note, we give some generalizations of Gödel's second incompleteness theorem and study their surroundings.

We revisit it from two perspectives.

One perspective is the relationship between the definable complexity of a theory and unprovability of its soundness.

We clarify the relationship between this perspective and induction axioms.

We also determine the logical strength of Craig's trick, which is important for studying the definability of a theory, from the point of view of reverse mathematics.

The other perspective is semantic incompleteness.

The second incompleteness theorem may be seen as the unprovability of the existence of a model.

It is known that `model' is replaced with `$\omega$-model' or `$\beta_n$-model'.

We give a new and unified proof of the $\omega$-model and $\beta_n$-model versions of the incompleteness theorem.