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A famous result of Goodstein asserts that the Goodstein sequence of integers terminates. For a precise statement and a short proof, see https://en.wikipedia.org/wiki/Goodstein%27s_theorem. A well ...

There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here. Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, ...

Gödel has shown that a consistent recursively axiomatizable first-order theory that can interpret Robinson arithmetic is incomplete. I think there is some sentimental value in working with a theory ...

Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...