Abstract Annals of Pure and Applied Logic 120 (2003) 1 – 63

n / Annals of Pure and Applied Logic 120 (2003) 1 – 63
gap ‘absolutely uninterpolatible’, and a summable gap (c.f. Deÿnition 2.1) satisÿes a considerably stronger additional requirement which has measure theoretic implications. In [35] Kunen proves that MAℵ1 implies that all (!1 ; !1 ) gaps are indestructible. On the other hand, it is a Theorem of TodorÄ cevià c (Theorem 1.10) that a destructible (!1 ; !1 ) gap can always be constructed from a Cohen real. This leads to the problem whether it is possible to randomize Kunen’s Theorem, a question which was apparently ÿrst considered by Woodin (see [62,13,5]). Question 1. (Woodin [60]). Does MAℵ1 imply that all (!1 ; !1 ) gaps are indestructible in any forcing extension by a measure algebra? In this paper we give a partial positive answer to Woodin’s question for the case of a separable measure algebra, i.e. in Theorem 3.20 we prove that MAℵ1 implies that all (!1 ; !1 ) gaps are indestructible in any forcing extension by a separable measure algebra. This is applied to gain deeper insights in Descriptive Set Theory. For example, we give an extension of the Harrington–Shelah Theorem [22] (see Theorem 4.1), by 1 showing that under MAℵ1 the Baire categoricity of all 1 3 non- 3 -complete sets of reals is enough to imply the existence of a weakly compact cardinal in the constructible universe. However, Question 1 in its full generality remains unsolved.
Annals of Pure and Applied Logic 120 (2003) 1 – 63
/locate/apal
Summable gaps
James Hirschorn
Department of Mathematics, University of Helsinki, Helsinki, Finland Received 3 May 2001; accepted 2 December 2001 Communicated by T. Jech
J. Hirschorn / Annals of Pure and Applied Logic 120 (2003) 1 – 63
3
In the case where the quasi-ordering is a Boolean algebra, a pre-gap is allowed to satisfy the following condition instead of (d): (d ∗ ) A ⊥ B, i.e. A · B = 0 for all A ∈ A and B ∈ B. Note that a ( ; Â) pre-gap (A; B) satisfying (d ∗ ) is equivalent to a ( ; Â∗ ) pre-gap satisfying (d) by considering (A; −B) where −B = {−B: B ∈ B}. (Recall that Â∗ is the reverse type of Â, e.g. (!1 ; ∈)∗ = (!1 ; ).) The primary objects of our study are gaps in the Boolean algebra (P(N)= ÿn; ⊆∗), in the sense of (d ∗ ). Recall that [A] ⊂∗ [B] i A\B is ÿnite and B\A is inÿnite (1.1)
Abstract It is proved, under Martin’s Axiom, that all (!1 ; !1 ) gaps in P(N)= ÿn are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin’s Axiom the Baire categoricity of all 1 3 non- 1 3 -complete sets of reals requires a weakly compact cardinal. c 2002 Elsevier Science B.V. All rights reserved.
1. Preliminaries First a word on notation. The usual variants of Martin’s Axiom are associated with cardinal characteristics. We follow Blass’ notation in [10]. Thus for a class C of posets, m(C) is equal to the smallest cardinal Ä such that Martin’s Axiom for Ä many dense sets fails for some poset in C. And m is a short form of m(ccc) (i.e. m is the smallest cardinal such that MAm fails). We begin with a review of the deÿnitions and basic results concerning gaps. Deÿnition 1.1. A quasi-ordering is a set with an ordering which is re exive and transitive. A linear ordering is a quasi-ordering which is totally ordered. And a partial ordering is a quasi-ordering which is irre exive, i.e. p6q and q6p implies p = q. By a type we mean an isomorphism type of some linear ordering. Deÿnition 1.2. Let (P; 6) be a quasi-ordering. For a pair of types ( ; Â), a pair (A; B) where A; B ⊆ P is called a ( ; Â) pre-gap in P if (a) A and B are linearly ordered by 6, (b) (A; 6) has type , (c) (B; 6) has type Â, (d) A6B, i.e. A6B for all A ∈ A and B ∈ B. The pre-gap is called a ( ; Â) gap in P if moreover (e) there is no C ∈ P such that A6C for all A ∈ A and B¿C for all B ∈ B. Such a condition C is said to interpolate A and B.
MSC: 03E05; 60C05; 03E15; 03E40 Keywords: Indestructible gap; Haar random real; Baire property of
1 3 1 -complete 3
non-
sets
0. Introduction One of the most striking early discoveries in Set Theory was Hausdor ’s construction of a gap inside the Boolean algebra P(N)= ÿn. And one of the more interesting phenomenon associated with these gaps is the phenomenon of destructibility, that is the possibility of interpolating a gap (and thus rendering it a non-gap) in some forcing extension (c.f. Deÿnitions 1.2 and 1.7). The question of whether a given (!1 ; !1 ) gap can be interpolated in some forcing extension is obviously a natural one in the context of studying Forcing, but in our context there are other reasons for asking this question. What we are really interested in here are additional properties which may be imposed on a given (!1 ; !1 ) gap, and the relationship between those properties and the possibility of interpolating the gap. For example, Hausdor ’s original construction of an (!1 ; !1 ) gap isolates a fairly strong additional property (c.f. (1.12)) which makes his