Coding without fine structure

Extract

In this paper we prove Jensen's Coding Theorem, assuming ˜ 0^#, via a proof that makes no use of the fine structure theory. We do need to quote Jensen's Covering Theorem, whose proof uses fine-structural ideas, but make no direct use of these ideas. The key to our proof is the use of “coding delays.”

Coding Theorem (Jensen). Suppose 〈M,A〉 is a model of ZFC + O^#does not exist. Then there is an 〈M, A〉-definable class forcing P such that if G ⊆ P is P-generic over 〈M, A〉:

(a) 〈M[G],A,G〉 ⊨ NZFC.

(b) M[G] ⊨ V = L[R], R ⊆ ωand 〈M[G], A, G〉 ⊨ A,G are definable from the parameter R.

In the above statement when we say “〈M, A〉 ⊨ ZFC” we mean that M ⊨ ZFC and in addition M satisfies replacement for formulas that mention A as a predicate. And “P-generic over 〈M, A〉” means that all 〈M, A〉-definable dense classes are met.

The consequence of ˜ O^# that we need follows directly from the Covering Theorem.