Determinacy from strong compactness of $ω_1$

Abstract

In the absence of the Axiom of Choice, the "small" cardinal ω 1 can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that ω 1 is Xstrongly compact (where X is any set) if there is a fine, countably complete measure on ℘ ω 1 (X). Working in ZF + DC, we prove that the ℘(ω 1 )-strong compactness and ℘(R)-strong compactness of ω 1 are equiconsistent with AD and AD R + DC respectively, where AD denotes the Axiom of Determinacy and AD R denotes the Axiom of Real Determinacy. The ℘(R)-supercompactness of ω 1 is shown to be slightly stronger than AD R + DC, but its consistency strength is not computed precisely. An equiconsistency result at the level of AD R without DC is also obtained.

On supercompactness of $ω_1$

“…We conjecture that HPC is not needed in Theorem 3. [17] has shown that ω 1 is supercompact implies that all sets in L(R) are Suslin co-Suslin and are determined and much more. 13 One may hope to prove Conjecture 3 by showing that every Suslin co-Suslin set is homogeneously Suslin.…”

“…Through work of Harrington, Kechris, Neeman, Woodin amongst others, we know that ω 1 is α-supercompact for every ordinal α < Θ under AD + , a strengthening of AD. By [17], AD and AD R cannot imply ω 1 is supercompact. Woodin (see below) shows that AD is consistent with "ω 1 is supercompact.…”

“…"ω 1 is supercompact" also implies a large collection of sets of reals are determined ( [17]) and perhaps an even larger collection of sets of reals admit ∞-Borel representations.…”