On Gödel’s second incompleteness theorem

In this note we give a short proof of Godel's Second Incompleteness Theorem. G6del's Second Incompleteness Theorem states that no sufficiently strong consistent mathematical theory can prove its own consistency [1]. In this note we give a short proof of the theorem. Theorem. It is unprovable in set theory (unless it is inconsistent) that there exists a model of set theory. Proof. Assume that set theory is consistent and that it proves that a model of set theory exists. Let I be a finite set of axioms sufficiently strong to formulate the concepts "model" and "satisfies" and to prove the existence of a model of set theory. For the rest of the proof, a model means a model of I and letters M and N denote models. If m is a set with a binary relation, Em denotes that relation. If N l= (E is a relation), then E* is the relation consisting of all pairs (x, y) such that N F xEy. If M and N, are models we define M < N if there exists some m E N such that EM= (Em). (Informally, M < N means that M is, in the real world, the structure that N thinks m is.) If M < N then for every sentence a (1) M F a if and only if N =(m = a). In particular, N F (m is a model). Also, if N l (m is a model), then (em) is the E-relation of some model M < N. It follows that (2) if M1 < M2 and M2 < M3 then M1 < M3. Let us consider some fixed coding of formulas by numbers (Godel numbering), and let S, be the name for the nth definable set of numbers. Received by the editors September 29, 1992 and, in revised form, November 30, 1992. 1991 Mathematics Subject Classification. Primary 03B99, 03C99, 03E99, 03F99. Supported in part by NSF grant DMS-8918299. ? 1994 American Mathematical Society 0002-9939/94 $1.00 +$.25 per page