Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

We determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

Canonical measure assignments

We work under the assumption of the Axiom of Determinacy and associate a measure to each cardinal κ < ℵ ε0 in a recursive definition of a canonical measure assignment. We give algorithmic applications of the existence of such a canonical measure assignment (computation of cofinalities, computation of the Kleinberg sequences associated to the normal ultrafilters on all projective ordinals).

Borel-amenable reducibilities for sets of reals

We show that if ℱ is any “well-behaved” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on (ωω) induced by ℱ turns out to look like the Wadge hierarchy (which is the special case where ℱ is the set of continuous functions).