Coding without Fine Structure

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〉 |= ZFC. (b) M [G] |= V = L[R], R ⊆ ω and 〈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.