PROPER FORCINGS AND ABSOLUTENESS IN L(R).pdf

PROPER FORCINGS AND ABSOLUTENESS IN L(R) Itay Neeman and Jindr?ich Zapletal Harvard University California Institute of Technology Abstract. We show that in the presence of large cardinals proper forcings do not change the theory of L(R) with real and ordinal parameters and do not code any set of ordinals into the reals unless that set has already been so coded in the ground model. 0. Introduction It is a well-established fact by now that in the presence of large cardinals the minimal model L(R) of ZF set theory containing all reals and ordinals has strong canonicity propeties–for example it satisfies the Axiom of Determinacy and its parameter-free theory is the same in all set generic extensions of the universe [MS, W1]. In this paper we give full proofs of three absoluteness theorems connecting the model L(R) with the basic forcing-theoretic notion of properness [Sh]. Embedding Theorem. Let δ be a weakly compact Woodin cardinal and P a proper forcing notion of size δ. Then in V P there is an elementary embedding j : L(RV ) → L(RV P ) which fixes all ordinals. This is related to the results of [FM, Theorem 3.4] and implies that in the presence of large cardinals proper forcings cannot change the ordinal parametrized theory of L(R), in particular, the values of the projective ordinals or θL(R). On the other hand, it is known that semiproper forcings can increase the value of δ12 [W2] and so the Embedding Theorem cannot be generalized to such posets. Anticoding Theorem. Let δ be a weakly compact Woodin cardinal, P a proper forcing notion of size δ and A ? Ord. Then A ∈ L(R) if an only if P ° A? ∈ L(R). Thus while proper forcings can add many new reals to the universe no old sets of ordinals can be coded by these reals. This should be contrasted with [BJW]. Again, a generalization to semiproper forcings fails as shown in Section 7. 1991 Mathematics Subject Classification. 03E55, 03E40. The second author acknowledges support from NSF grant DMS 9022140, GA C?R grant