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Σ -Absoluteness in Forcing Extensions

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Σ -Absoluteness in Forcing Extensions 1 Σ3-Absoluteness in Forcing Extensions David Schrittesser June 2004 Diplomarbeit eingereicht zur Erlangung des akademischen Grades Magister der Naturwissenschaften an der Fakult¨atf¨urNaturwissenschaften und Mathematik der Universi¨atWien Betreuer: Sy Friedman Abstract 1 We investigate the consistency strength of the forcing axiom for Σ3 formulas, for various classes of forcings. We review that the consistency strength of 1 Σ3-absoluteness for all set forcing or even just for !1-preserving forcing is that of a reflecting cardinal. To get the same strength from the forcing axiom restricted to proper forcing, one can add the hypotheses that !1 is inaccessible to reals. Then we investigate the strength of the forcing axiom restricted to ccc forcing notions under this additional hypothesis; to gauge it we introduce a weak version of a weak compact cardinal, namely, a lightface 1 Σ2-indescribable cardinal. Contents 1 Preliminary facts 2 1.1 General set theory . 3 1.2 Forcing . 5 1.3 Reals . 14 1.4 Large cardinals . 18 2 Equiconsistency for set forcing 22 3 A lower bound for !1-preserving set-forcing 24 4 Coding and reshaping when !1 is inaccessible to reals 26 4.1 Preserving !1 ........................... 26 4.2 Preserving stationary subsets of !1 . 33 4.3 Proper forcing . 35 5 Forcing with the countable chain condition 39 1 5.1 Lightface Σ2-indescribable cardinals . 39 5.2 Coding using an Aronszajn-tree . 41 5.3 An equiconsistency . 43 1 1 Preliminary facts This section contains some definitions, well known facts and basic theorems which we will use throughout the present paper. First, we take a little time to fix our notation (we hope to include all M possible sources of confusion). For a structure M = hM; 2; P0;:::;Pki, L M denotes the language L = f2;P0;:::;Pkg. LX denotes the same language with constants for all the elements of X. We write M ≺ N to denote M elementarity (i.e. M and N have the same theory in LM ), and we write 00 π : M!Σ! N to express π is an elementary embedding (i.e. hran(π); π 2 00 00 ; π P0; : : : ; π Pki ≺ N ). For transitive models, we call the least ordinal α 2 M such that π(α) > α the critical point of π, and denote it by crit(π). For a structure N = hN; E; : : :i such that E is extensional and well-founded (and, in the case of a class sized structure, set-like), we denote its transitive collapse (that is, the unique structure hM; 2;:::i isomorphic to N and such that M is transitive) by tcoll(N ). By (φ)M we mean the formula φ holds relativized to a (possibly class-sized) structure M and by tM we mean the interpretation of a term t as from the viewpoint of M. When we talk about structures M = hM; 2; P0;:::;Pki, we sometimes omit listing the 2 as a predicate. When we use notation that is defined for structures in a context when we talk about sets M, it is implied we mean the structure hM; 2i. We denote the usual Zermelo-Fraenkel axioms by ZF , and that theory with AC (the axiom of choice) added by ZFC. ZF − and ZFC− denote the corresponding theories with the power set axiom deleted. When we talk about Skolem functions Fφ for a (possibly class-sized) struc- ture M, we consider M to be equipped with a well-ordering <M. Then, if M φ(x0; x1) is a formula of L with two free variables, for p 2 M, we define Fφ(p) to be the <M-least y 2 M such that M φ(y; p) if such y exists, and M to be ; otherwise. For X ⊆ M, we denote by hΣ! (X) the Skolem hull of X in N , that is the smallest (in the sense of ⊆) set Y ⊆ M s.t. Y ⊇ X that is M closed under all Skolem functions Fφ, for φ a formula of L . We use the familiar Σk; Πk; ∆k notation for the classes formulas that can be written with k blocks of equal quantifiers starting with 9 or 8 and for the intersection of these first two classes, respectively. We shall give a more thorough review of the analytical hierarchy in section 1.3; in section 5.1 we k quickly review the general Σn hierarchies of higher order formulas over a structure. We write Lim, Card, Reg for the class of limit ordinals, cardinals and regular cardinals respectively. For a set x, jxj is its cardinality, and we sometimes write x . y to mean that jxj < jyj. TC(x) denotes the transitive closure of x. For any cardinal κ, let Hκ denote the set of all x such that 2 TC(x) has size less than κ (i.e., the Hκ hierarchy is continuously defined also at singular cardinals). HC denotes H!1 , the set of hereditarily countable sets. As far as forcing is concerned, we sometimes write V P for a generic ex- tension V [G], where G is P -generic over V , if the particular G concerned is of no importance. We write dots over names (e.g.q _) andq _[G] for the interpretation of the nameq _ by a generic G. For sets a in the ground model we writea ˇ for the \standard name" for a. 1.1 General set theory 1.1 Definition. 1. C ⊂ [A]! is called cub if, and only if, there are predi- cates P1;:::;Pk ⊂ A such that ! C = fX 2 [A] j hX; P1;:::;Pki ≺ hA; P1;:::;Pkig 2. S ⊂ [A]! is called stationary if, and only if, for all cub sets C ⊂ [A]!, S \ C 6= ; 1.2 Fact. Let N = hN; R0;:::Rki, X 2 Hκ and X ⊆ N. Then there exists 0 0 a transitive M 2 Hκ s.t. X ⊆ M together with R0;:::Rk ⊆ M, and an 0 0 elementary embedding i : hM; R0;:::Rki !Σ! hN; R0;:::Rki s.t. i is the identity on X. Proof. First observe that since X 2 Hκ, also TC(X) 2 Hκ, whence we can assume X to be transitive. Now start by taking the Skolem-hull (with respect to the language where the additional predicates have been added): ¯ N ¯ ¯ let M := hΣ (X). M has size less than κ. Set M := tcoll(M), the transitive !¯ collapse of M. Being transitive and of size less than κ, M 2 Hκ. Let i be the inverse of the collapsing map (the new predicates are just the pullbacks of the original predicates). As X ⊆ M¯ was transitive, i is the identity on X. M 1.3 Fact. If N is transitive and λ + 1 ⊆ M ≺ N, then (Hλ+ ) is transitive. M Proof. Let x 2 (Hλ+ ) be arbitrary, we show x ⊆ M. M \9f : λ TC(x)", and so by elementarity, for some f 2 M, N \f : λ TC(x)". Now for arbitrary y 2 x, we have y = f(ξ) for some ξ 2 λ, while λ ⊆ M. It is easily checked that f; ξ 2 M by elementarity implies f(ξ) 2 M, so y 2 M. 1.4 Corollary. Let A be any set of ordinals, κ a successor cardinal. Then C := f α < κ j Lα[A] ≺ Lκ[A] g is cub in κ. 3 Proof. For ξ < κ, let χ(ξ) denote the least ordinal such that Skolem functions for Lκ[A] on Lξ[A] have values in Lχ(ξ)[A]. Then χ : κ ! κ and C is precisely the set of closure points of χ, hence cub. − 1.5 Fact. If M ZF is transitive, π : M !Σ! N an elementary embedding M M with critical point α, then α 2 Reg and π Hα = id. Proof. Assume M thinks α is singular; then there is a function f in M, with domain an ordinal below α and values below α whose range is cofinal in α. But as all these ordinals are not moved by π, π(f) = f and thus α = sup(ran(f)) = sup(ran(π(f))) = π(sup(ran(f))) = π(α), contradiction. As π α = id, α ⊆ ran(π). By hypothesis ran(π) ≺ N, we can apply 1.3: ran(π) ran(π) 00 M for each λ < α,(Hλ+ ) is transitive. So (Hπ(α)) = π (Hα) (as the union of these transitive sets) is seen to be transitive. Note that as the transitive collapse of any set A is uniquely defined as the transitive set isomorphic to A, we have that M must be the transitive collapse of ran(π) and π must be the inverse of the collapsing map. As the collapsing map is M just the identity on any transitive set, π is the identity on (Hα) . 1.6 Fact. Let M = hM; R1;:::;Rki, N = hN; S1;:::;Sri, be models such that k ≤ r, Ri has the same arity as Si for i ≤ k, and M is countable. Then the statement 9π; 9Rk+1;:::;Rr π : hM; R1;:::;Rri !Σ! hN; S1;:::;Sri (1) is absolute for transitive models U of ZF − such that U contains both N and ∼ M and U M = !. Proof. Let m : ! ! M be an enumeration of M,(φi)i2! an enumeration of formulas of the language of N . Let γ : ! ! !, δ : ! ! <!! be such N that together, they enumerate all formulas of the language LM (the language of N with constants for all the elements of M) in the following sense: for all n, ran(δ(n)) ⊆ n, the number of free variables of φγ(n) is no larger than dom(δ(n)), and γ and δ are onto in the sense that φγ(n)(m(s(0)); : : : ; m(s(l))), N for s = δ(n) and l = lh(s) − 1, runs through all formulas of LM as n runs through !.
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