A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page

Indestructibility of Vopenka's Principle

We show that Vopenka's Principle and Vopenka cardinals are indestructible under reverse Easton forcing iterations of increasingly directed-closed partial orders, without the need for any preparatory forcing. As a consequence, we are able to prove the relative consistency of these large cardinal axioms with a variety of statements known to be independent of ZFC, such as the generalised continuum hypothesis, the existence of a definable well-order of the universe, and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic

The nonexistence of robust codes for subsets of ω

Several results are presented concerning the existence or nonexistence, for a subset S of ? , of a real r which works as a robust code for S with respect to a given sequence of pairwise disjoint stationary subsets of ? , where "robustness" of r as a code may either mean that S ? L[r, ] whenever each S * is equal to S modulo nonstationary changes, or may have the weaker meaning that S ? L[r, ] for every club C ? ? . Variants of the above theme are also considered which result when the requirement that S gets exactly coded is replaced by the weaker requirement that some set is coded which is equal to S up to a club, and when sequences of stationary sets are replaced by decoding devices possibly carrying more information (like functions from ? into ? )

On simple partitions of [κ]

For every uncountable regular cardinal ?, every ?-Borel partition of the space of all members of [?] whose enumerating function does not have fixed points has a homogeneous club

A maximal bounded forcing axiom

After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets G such that, letting G be the class of all stationary-set-preserving partially ordered sets, one can prove the following: (a) G ? G, (b) G = G if and only if NS is N-dense. (c) If P ? G, then BFA({P}) fails. We call the bounded forcing axiom for G Maximal Bounded Forcing Axiom (MBFA). Finally we prove MBFA consistent relative to the consistency of an inaccessible S-correct cardinal which is a limit of strongly compact cardinals

On a convenient property about [γ]

Several situations are presented in which there is an ordinal ? such that is a stationary subset of for all stationary subset of [?] for all stationary S,T ??. A natural strengthening of the existence of an ordinal ? for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of and the existence of sharps for all reals. Also, an optimal model separating Bounded Semiproper Forcing Axiom (BSPFA) and Bounded Martin's Maximum (BMM) is produced and it is shown that a strong form of BMM involving only parameters from Himplies that every function from ? into ? is bounded on a club by a canonical function

Bounded Martin's Maximum, Weak ErdoS cardinals, and ψ

We prove that a form of the Erdos property (consistent with V = L[H] and strictly weaker than the Weak Chang's Conjecture at ?), together with Bounded Martin's Maximum implies that Woodin's principle ? holds, and therefore 2 = ?. We also prove that ? implies that every function f: ? ? ? is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails