A Proof of Projective Determinacy

Abstract

Primary 03E 15, 03E55; Secondary 03E60.The authors would like to give special thanks to the referees, whose lists of corrections and suggestions were extraordinary in both length and quality.

Questions on generalised Baire spaces

“…It is known that universally Baire sets of reals are exactly the “∞‐homogeneously Suslin sets of reals” under suitable large cardinals (cf. for more on ∞‐homogeneously Suslin sets). The third question concerns a characterisation of sets in terms of homogeneously Suslin sets.…”

The well-foundedness of the Mitchell order

“…?1 and 3 of [MS1], and in ? As we shall state and prove our main result at this level of generality, we record some basic definitions and facts here.…”

“…Notice that we do not assume A < j(K), or equivalently a(E) = K + 1, as is assumed in [MS1] and [MS2]. We set crit(E) = K and lh(E) = A.…”

“…Our concept of an iteration tree is more general in several ways than that of [MS1] ? lh(E) I 2crit(E) follows that str(E) < I lh(E)I 2crit(E).…”

Perception, Intuition, and Reliability

“…By contrast, the aforementioned derivation of PD from infinitely many Woodin cardinals represents a proof that is fulfilling in the proper sense. For Martin and Steel's (, p. 74) point of departure had been the hope that an obstacle to building canonical models for those cardinals may be turned into a method for proving determinacy. Their argument confirmed this expectation by exhibiting a mechanism for propagating certain representations for sets of reals through the projective hierarchy – and thereby obtaining projective determinacy – via the same kind of structures that generate complicated wellorders of the real numbers in canonical models with Woodin cardinals…”