On Gödel’s Second Incompleteness Theorem

Gödel’s Second Incompleteness Theorem states that no sufficiently strong consistent mathematical theory can prove its own consistency. In [1] this is proved for every axiomatic theory extending the Peano Arithmetic. For axiomatic set theory a simpler proof was given in [2] using the fact that in set theory, consistency of a set of axioms is equivalent to the existence of a model. In this note we give a very simple proof of Gödel’s Theorem for set theory: Theorem. It is unprovable in set theory (unless it is inconsistent) that there exists a model of set theory. By “set theory” we mean any axiomatic set theory with finitely many axioms. The proof (suitably modified) works for any theory sufficiently strong to formulate the concepts “model”, “satisfies” and “isomorphic”, such as second order arithmetic and its weaker versions. If M and N are models of set theory (henceforth models), we define