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Forcing Axioms and Projective Sets of Reals FORCING AXIOMS AND PROJECTIVE SETS OF REALS RALF SCHINDLER Abstract. This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + \every projective set of reals is Lebesgue measurable" is equiconsistent with ZFC + \there is a Σ1 reflecting cardinal above a remarkable cardinal." 1. Introduction. The current paper∗ is in the tradition of the following result. Theorem 1. ([HaSh85, Theorem B]) Equiconsistent are: (1) ZFC + there is a weakly compact cardinal, and (2) ZFC + Martin's axiom (MA) + every projective set of reals is Lebesgue measurable. This theorem links a large cardinal concept with a forcing axiom (MA) and a regularity property for the projective sets of reals. We shall be interested in considering forcing axioms which are somewhat stronger than MA. In particular, we shall produce the following new result.y Theorem 2. Equiconsistent are: (1) ZFC + there is a Σ1 reflecting cardinal above a remarkable car- dinal, and (2) ZFC + the bounded proper forcing axiom (BPFA) + every projec- tive set of reals is Lebesgue measurable. 1991 Mathematics Subject Classification. Primary 03E55, 03E15. Secondary 03E35, 03E60. Key words and phrases. set theory/forcing axioms/large cardinals/descriptive set theory. ∗The author would like to thank Ilijas Farah, Ernest Schimmerling, and Boban Velickovic for sharing with him mathematical and historical insights. yThis theorem was first presented in a talk at the 6e Atelier International de Th´eorie des Ensembles in Luminy, France, on Sept 20, 2000. 1 2 RALF SCHINDLER We don't expect the reader to be familiar with the concepts appear- ing in the previous theorems, as we are going to explain them here. We do have to presuppose, though, a basic understanding of forcing and constructibility theory; this material is covered by the text books [Je78] and [Ku80]. Large cardinals are ubiquitous in set theory. In [Hau14], Hausdorff had asked whether there is a regular fixed point of the aleph function i 7! @i. We know today that the existence of such a fixed point cannot be shown in ZFC, because if κ is regular and κ = @κ then Lκ j= ZFC. (Here and in what follows we assume that ZFC is consistent.) We may say that a formula φ(−) defines a large cardinal concept if ZFC proves that 8κ(φ(κ) ) Lκ j= ZFC). A large cardinal axiom is a statement of the form 9κ φ(κ) for some large cardinal concept given by φ(−). Any large cardinal axiom is thereby independent from ZFC. Forcing axioms are also independent from ZFC. The prototype of all forcing axioms, Martin's axiom, was isolated from the Solovay- Tennenbaum proof of the consistency of the non-existence of Suslin trees on !1 (cf. [Je78, Theorem 50] or [Ku80, II, x4]). A forcing axiom asserts the existence of reasonably generic filters for posets from a fixed class of forcings. One may often view forcing axioms as a substitute for CH (the continuum hypothesis) in contexts where CH fails. Projective sets of reals are primarily studied in descriptive set the- ory. A projective set comes from a closed set via a finite process of taking complements and projections (cf. [Je78, p. 500]). ZFC typically decides questions about \simple" projective sets, but it leaves the ar- bitrary case open. For instance, every analytic set of reals is Lebesgue 1 measurable in ZFC, whereas the Lebesgue measurability of all ∆2 sets of reals is independent from ZFC. Nevertheless, most descriptive set theorists believe it is true that every projective set of reals is Lebesgue measurable. We are now going to study the Lebesgue measurability of all projec- tive sets of reals in the context of forcing axioms. Set theorists obtain insight into the mathematical structure provided by various hypotheses by determining their consistency strength. We'll write Con(Σ) ) Con(Σ0) for the statement that the consistency of the theory Σ implies that of Σ0. We say that Σ and Σ0 are equiconsistent just in case Con(Σ) , Con(Σ0). The consistency strength (or, large cardinal strength) of a given hypothesis Ψ is that large cardinal axiom 9κ φ(κ) such that ZFC + Ψ is equiconsistent with ZFC + 9κ φ(κ). It is an empirical fact (and one of the miracles of set theory) that \natural" hypotheses do have a consistency strength; this paper in fact FORCING AXIOMS AND PROJECTIVE SETS OF REALS 3 gives examples. We refer the reader to [Ka94] for extensive information about this phenomenon. The current paper can also be read as an introduction to a method for proving equiconsistency results. Typically, the direction Con(ZFC + Ψ) ) Con(ZFC + 9κ φ(κ)) uses constructibility theory, whereas Con(ZFC + 9κ φ(κ)) ) Con(ZFC + Ψ) is an application of the method of forcing. 2. MA and the Harrington-Shelah theorem. Let P be a poset, and let D be a collection of sets such that every D 2 D is a subset of P. We say that a filter F ⊂ P is D-generic just in case F \ D =6 ; for every D 2 D. D ⊂ P is predense (in P) if every p 2 P is compatible with some q 2 D. D ⊂ P is dense (in P) if for every p 2 P there is some stronger q 2 D. If D is the collection of all dense subsets of P then a D-generic filter cannot exist, unless P contains atoms. On the other hand, if D is countable then a D-generic filter always exists. In particular, if M is a countable transitive model of ZFC with P 2 M and if D 2 M is the collection of all the dense subsets of P which exist in M then there is a D-generic filter; it is one of the key building blocks of the theory of forcing that such a filter G can then be \adjoined" to M to produce another model of ZFC, M[G]. The approach of forcing axioms is to ask for more than countably many dense sets to be met at once, where P will have to belong to a specific class of posets. In what follows we shall focus on the case that no more than @1 many dense sets are to be met. Definition 3. Let Γ ⊂ V be a class of posets. We say that MA(Γ) holds if for every P 2 Γ and for every collection D of sets such that • Card(D) ≤ @1, and • D is dense in P for every D 2 D there is some D-generic filter. Let c.c.c. denote the class of all posets which have the countable chain condition. We take Martin's Axiom, which is abbreviated by MA and which was mentioned in Theorem 1, to say that MA(c:c:c:) @0 holds. In particular, MA { as we define it { implies that 2 ≥ @2 (i.e., that the continuum hypothesis, CH, is false). The reader may consult [Je78, xx22, 23, 44] or [Ku80, Chapters II and VII] on MA. One can construe MA as a generalization of the Baire category theorem (cf. [Ku80, Theorem II.2.22]). The reader might also enjoy the discussion of the plenty consequences of MA that can be found in [Fr84]. 4 RALF SCHINDLER It can be shown that Con(ZFC) ) Con(ZFC + MA) (cf. the para- graph right after Theorem 5 below). It is actually consistent to require that κ many dense sets can be met for any κ < 2@0 , but it is inconsistent to require 2@0 many dense sets to be met. By [Ku80, Theorem II.3.4], MA is equivalent to the fact that for every P 2 c:c:c: which is of size ≤ @1 and for every collection D of sets such that Card(D) ≤ @1 and D is dense in P for every D 2 D there is some D-generic filter. This readily implies that MA is equivalent to BMA(c:c:c:) according to the following definition. Definition 4. Let Γ ⊂ V be a class of posets. We say that BMA(Γ) holds if for every P 2 Γ and for every collection D of sets such that • Card(D) ≤ @1, and • D is predense in P and has size ≤ @1 for every D 2 D there is some D-generic filter. In order to uncover the relation of MA with the Lebesgue measura- bility of projective sets of reals we need a first large cardinal concept. We say that a cardinal κ is inaccessible if κ > @0, κ is regular, and γ 2 < κ for all cardinals γ < κ. We say that !1 is inaccessible to L[x] the reals if !1 is countable for every real x. Notice that if !1 is inaccessible to the reals then !1 is an inaccessible cardinal from the point of view of L. Theorem 5. ([Sh84]) If every projective set of reals is Lebesgue mea- 1 surable then !1 is inaccessible to the reals. In fact, if every Σ3 set of reals is Lebesgue measurable then !1 is inaccessible to the reals. There is a poset P 2 c:c:c: such that MA provably holds in V (cf. [Je78, x23] or [Ku80, VIII. x6]). Let us assume that V = L. We ¡ shall then have that !V = !V = !L, and we may apply Theorem 5 1 1 1 inside V to get that in V there is a projective set of reals which is not Lebesgue measurable.
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