Supercompact cardinals, sets of reals, and weakly homogeneous trees

Abstract

It is shown that if there exists a supercompact cardinal then every set of reals, which is an element of L(R), is the projection of a weakly homogeneous tree. As a consequence of this theorem and recent work of Martin and Steel [Martin, D. A. & Steel, J. R. (1988) Proc. Nall. Acad. Sci. USA 85, 6582-65861, it follows that (if there is a supercompact cardinal) every set of reals in L(R) is determined.The subtle relationships between the existence of certain large cardinals and regularity properties of various simple sets of reals is one of the striking developments in modem set theory.One of the first results in this direction is that of R. I show here that the existence of a supercompact cardinal implies that every set of reals that belongs to L(R) has a certain structural representation from which the regularity results, such as measurability, easily follow. This is made more precise through a sequence of definitions.For our purposes the set of reals, R, is the set 0d of all functions f:o -t) , wherew = {0, 1, . . ., k, . . .}is the set of nonnegative integers. We let I'm denote the set of all finite sequences of elements of w and for s E i< 'let N, be the set, Ns = {f 8 a': f [ 1(s) = s}, where 1(s) = length(s). The set {N, : s E A<"} generates a topology on of it is the product topology derived from the discrete topology on 0. Endowed with this topology a'xis homeomorphic to the Euclidean space of irrationals. Suppose X is a set. We denote by X 'the set of all functions f: t-s Xand we denote byX<' the set of all finite sequences of elements ofX. We adopt the usual convention that X"' is the set of all functions f: dom f xsuch that dom f Ecowand ifs E X" then dom s = i(s) = length(s). Suppose A is an ordinal, A > 0. A tree on o x A is a subset T C a' x AX"' ' such that for all pairs (s, t) E T, Suppose X is a nonempty set. We denote by m(X) the set of countably complete ultrafilters on the Boolean -algebra P(X). , is a measure on X if ,u E m(X-). For A E m(X) and A C X we write ,u(A) = 1 to indicate A E Au. Suppose that X = Y<' and that ,t E m(Y< '). Since A is countably additive, there is a unique k E w such that IL(yk) = 1. Suppose .ul, y, E m(Y< '), ,ut(Y') = 1, and ,2(Ykl) = 1. Then A < 2 (u2 projects to ,ul) if k1 < k2 and, for all A C yki, p 1(A) = 1 if and only if 1,2(A*) = 1 where A* = {s E Yk: s t ki E A}. such that for all k E w, (x rk, yrk) E domirand (ir(xrk, y [ k) : k E a) is a countably complete tower. I(s) = I(t) and (s t i, t [ i) E Tfor all i < I(s), i 6587The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

The method of forcing

“…See also Mathias's infinite dimensional generalization of Ramsey's Theorem which holds in L(R) after collapsing an appropriate large cardinal [19]. An explanation of the special role Solovay's model plays in the foundations of mathematics can be found in [23] [34]. [17] provides a good introduction to the methods needed to establish absoluteness results about L(R).…”

Proof of a Conjecture of Galvin

“…The countable stationary tower will be our main tool for proving Theorem 11. Building on the groundbreaking work of Foreman, Magidor, and Shelah [8], Woodin introduced the stationary tower in [27] and established a wide variety of results in set theory with it. Larson [15] provides an excellent and accessible introduction to the stationary tower and its applications.…”

“…The conditions on X which allow us (using large cardinals) such transfer to a Baire space Z and a continuous nowhere constant map f had been already used in the paper [23], which in turn was motivated by a problem of Haydon [11] from the theory of differentiability in the context of general Banach spaces. It should also be noted that large cardinals are introduced into the construction of Z and f : Z → X through the ideas behind the stationary tower forcing of Woodin [27], which in turn was inspired by the groundbreaking work of Foreman, Magidor and Shelah [8]. We believe that applying large cardinals to structural Ramsey theory is a new idea that will give us more results of this kind.…”

The Prospects for Mathematical Logic in the Twenty-First Century

“…This theory also fits beautifully within the framework of global set theory as currently developed through the theory of large cardinals. The determinacy principle is in fact "equivalent", in an appropriate sense, to the existence of certain types of large cardinals (see , Woodin [67]). Moreover the structure theory of definable sets in Polish spaces, that determinacy unveils, has a very close and deep relationship with the unfolding inner model theory of large cardinals, an example of which was so vividly illustrated in Itay Neeman's talk in this conference.…”