Proper Forcing and Remarkable Cardinals II

The current paper proves the results announced in [6]. We isolate a new large cardinal concept, “remarkability.” Consistencywise, remarkable cardinals are between ineffable and ω-Erdös cardinals. They are characterized by the existence of “0-like” embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness for proper forcings. In particular, said absoluteness does not imply Π1 determinacy. Large cardinals are widely used for measuring the consistency strength of set theoretic principles. The current paper isolates a new large cardinal concept, “remarkability,” which measures the power of proper forcing to change (in a certain respect) the shape of the universe. In fact, in this paper we shall give proofs of the main result announced in [6]. Let F be a class of set-sized posets. We say that L(R) is absolute for forcings of type F if for all P ∈ F , for all G being P -generic over V , for all formulae Φ(~v), and for all ~x ∈ R do we have that L(R ) |= Φ(~x) ⇔ L(R ) |= Φ(~x). We say that L(R) is absolute for c.c.c. (or, proper, ..., set) forcing if L(R) is absolute for forcings of type F where F = {P :P has the c.c.c.} (or, F = {P :P is proper}, ..., F = {P :P is any poset (in V )}). The existence of large cardinals (for example, of a proper class of Woodin cardinals) implies that L(R) is absolute for set forcing. (This is due Woodin.) The upshot is that an L(R)-stability of this sort even proves that AD, the Axiom of Determinacy, holds in L(R). (This is due to Woodin; a slightly weaker version of it was shown later and independently by Steel; cf. [9].) However, L(R) absoluteness for forcings of type F can be considerably weaker than AD if F is sufficiently ∗The author is indebted to Joan Bagaria, Sy Friedman, and Philip Welch for stimulating hints and observations. 1991 Mathematics Subject Classification. Primary 03E55, 03E15. Secondary 03E35, 03E60.