Combined Maximality Principles up to large cardinals

Abstract

The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high. As a by-product, assuming the consistency of a supercompact cardinal, I show that it is consistent that the least weakly compact cardinal is indestructible.

Generic embeddings associated to an indestructibly weakly compact cardinal

“…This gives the answer to a question I had at one point: The answer is no, since it is consistent that the least weakly compact cardinal κ is indestructible (see [7,Thm. 3.11]).…”

“…i.e., there is a g : P κ (λ) N −→ N in N such that for any f : 7. For X ∈ P (P κ (λ)) N , X ∈ F ⇐⇒ j''λ ∈ j(X ).…”

“…. , x l−1 )) 7. This makes sense because if there is an X such that a is stationary in X , then X = ∪a.…”

The envelope of a pointclass under a local determinacy hypothesis

“…The argument is also similar to that used by Kechris In order to state the lemma we first make a definition. For a set of reals A and a Turing ideal I , 3 we say that A is determined on I if there is a strategy in I (more precisely, coded by a real in I ) for the game G A that defeats every real in I . In other words, there is a strategy σ ∈ I such that for all reals x, y ∈ I we have σ * y ∈ A (if σ is a strategy for Player I) and x * σ / ∈ A (if σ is a strategy for Player II).…”

“…3 A Turing ideal is a set of reals that is closed under recursive join ⊕ and is downward closed under Turing reducibility ≤ T .…”