Delta1-Definability

Boban Veli ckovi c y Jensen's Coding Theorem provides a method for making a class of ordinals 2-deenable in a real. In this paper we explore the possibility of obtaining a 1-deenition. While this is not possible in general, we show that by assuming enough condensation, a class can be made 1 in a real parameter in a class-generic extension. Our main application is the following. Theorem There is a real R class-generic over L such that L ?cof ! = f : L ?cof() = !g is 1 over LR] in the parameter R. Every L-cardinal, with the exception of @ L 1 remains a cardinal in LR]. Moreover the existence of such a real R follows from the existence of O #. Every L-deenable class is 1-deenable over LO # ]. The following result shows that conversely if every L-deenable class is 1-deenable in a real parameter than O # exists. Proposition 1 Suppose L-card = f : is a cardinal in Lg is 1 in a real parameter. Then O # exists. PROOF: @ !+1 2 L-card so if L-card were 1 in a real, by reeection there would be unboundedly many < @ !+1 also in L-card. So @ !+1 > (@ !) +L and by Jensen's Covering Theorem, O # exists. a An L-amenable class A is 1-complete if every 1 (L) class is 1 hL;Ai.