Proper forcing extensions and Solovay models - CiteSeerX
Archive for Mathematical Logic manuscript No. (will be inserted by the editor)
Joan Bagaria1 · Roger Bosch2
Proper forcing extensions and Solovay models c Springer-Verlag 2003 Received: date / Revised version: date – ° Abstract. We study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every Σ 13 strongly-proper forcing notion ∼ preserves this property. This yields that the consistency strength of the absoluteness of 1 L(R) under Σ 3 strongly-proper forcing notions is that of the existence of an inaccessible ∼ cardinal. Further, the absoluteness of L(R) under projective strongly-proper forcing notions is consistent relative to the existence of a Σ ω -Mahlo cardinal. We also show that the ∼ consistency strength of the absoluteness of L(R) under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.
1. Introduction A Solovay model is, essentially, the model discovered by Solovay in [12], in which every set of reals is Lebesgue measurable, has the property of Baire, has the perfect set property, etc. That is, if κ is an inaccessible cardinal in some model V , then the L(R) of a model M resulting from collapsing κ to ω1 over V using the Levy collapse Coll(ω, < κ) is a Solovay model over V . In this situation, M has the following properties: 1. For every x ∈ R, ω1 is an inaccessible cardinal in V [x]. 2. Every x ∈ R is small-generic over V . That is, there is a forcing notion P in V , which is countable in M , and there is, in M , a P-generic filter g over V such that x ∈ V [g]. Woodin has showed (see [4]) that the converse is essentially true. Namely, Lemma 1. Suppose that V ⊆ M are models of (a fragment of ) ZFC and M satisfies (1) and (2) above. Then there is a forcing notion W in M which does not add new reals and creates a Coll (ω, < ω1 )-generic filter C over V such that M and V [C] have the same reals. 1: Instituci´ o Catalana de Recerca i Estudis Avan¸cats (ICREA) and Departament de L` ogica, Hist` oria i Filosofia de la Ci`encia. Universitat de Barcelona. Baldiri Reixac, s/n. 08028 Barcelona, Catalonia (Spain) e-mail: [email protected] 2: Departamento de Filosof´ıa. Universidad de Oviedo. Teniente Alfonso Mart´ınez, s/n. 33011 Oviedo, Asturias (Spain) e-mail: [email protected] Research partially supported by the research projects BFM2002-03236 of the Spanish Ministry of Science and Technology, and 2002SGR 00126 of the Generalitat de Catalunya. The second author was also partially supported by the research project GE01/HUM10, Grupos de excelencia, Principado de Asturias. Key words or phrases: Solovay models – generic absoluteness – strongly-proper forcing – projective forcing – consistency strength – definably-Mahlo cardinals – Mahlo cardinals – weakly-compact cardinals.
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Joan Bagaria, Roger Bosch
The conditions of W are filters which are generic over V for some initial segment of the Levy-collapse of ω1M . Thus, W forces that L(R)M is a Solovay model over V . So, in view of this we define: Definition 1. L(R)M is a Solovay model over V ⊆ M iff M satisfies: 1. For every x ∈ R, ω1 is an inaccessible cardinal in V [x] and 2. Every x ∈ R is small-generic over V . Our interest of the preservation of the property of being a Solovay model under forcing extensions that do not collapse ω1 lies mainly in the fact that it implies a strong form of generic absoluteness, which in turn has strong consequences for the theory of the reals (see the forthcomming [2]). By generic absoluteness we mean the following: Definition 2. Let V be a model of ZF . Let P ∈ V be a forcing notion and let ϕ be a formula (possibly with parameters in V ). V is ϕ-absolute for P iff V |= ϕ iff V P |= ϕ. If Σ is a set of formulas, V is Σ-absolute for P iff for every ϕ ∈ Σ, V is ϕ-absolute for P. Given a class of posets Γ , V is Σ-absolute for Γ iff for every P ∈ Γ , V is Σ-absolute for P in V . V is L(R)-absolute for P iff there exists an elementary embedding j : L(R)V → L(R)V
P
that fixes all the ordinals (and, of course, all the reals). For Γ a class of posets, V is L(R)-absolute for Γ if it is L(R)-absolute for every P in Γ . The main fact connecting Solovay models and generic absoluteness is given by the following lemma (see [4]). Lemma 2. Suppose that L(R)M and L(R)N are Solovay models over V such that RM ⊆ RN and ω1M = ω1N . Then there exists an elementary embedding j : L(R)M → L(R)N which fixes all the ordinals. An even stronger form of generic absoluteness is the following (see [4]): ˙ a P-name for a forcing notion. Definition 3. Let P be a forcing notion in V , and Q ˙ V is two-step ϕ-absolute for P and Q , if ˙
V P |= ϕ iff V P∗Q |= ϕ. (Note that ϕ may have parameters in V P , not just in V .) For Σ a set of formulas, ˙ in the obvious way. For Γ a class we define V is two-step Σ-absolute for P and Q ˙ of posets, V is two-step Σ-absolute for Γ iff for every P ∈ Γ and every P-name Q ˙ for a forcing notion in Γ , Σ is absolute for P and Q. ˙ iff there exists an elementary embedding V is L(R)-two-step absolute for P and Q P
j : L(R)V → L(R)V
˙ P∗Q
that fixes all the ordinals. For Γ a class of posets, V is L(R)-two-step absolute for Γ iff for every P ∈ Γ ˙ for a forcing notion in Γ , V is L(R)-two-step absolute for P and every P-name Q ˙ and Q.
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This paper continues the work started in [4] where we proved several equiconsistency results on generic absoluteness for projective ccc posets. More precisely, Definition 4. A Σ 1n poset (n ≥ 1) is a triple P = hP, ≤P , ⊥P i, where ≤P is a Σ 1n ∼ ∼ subset of ω ω × ω ω , P = field(≤P ), hP, ≤P i is a partial order, and ⊥P is a Σ 1n subset ∼ of ω ω × ω ω contained in P × P such that for every x, y ∈ P , x ⊥P y iff for no z ∈ P we have that z ≤P x and z ≤P y; i.e., iff x, y are incompatible. Similarly, we define Π 1n posets by substituting Π 1n for Σ 1n in the above definition. A ∆1n poset is a poset ∼ ∼ ∼ ∼ that is both Σ 1n and Π 1n . Finally, P is a projective poset iff P is Σ 1n , for some n. ∼ ∼ ∼ 1 1 A ccc poset P is strongly-Σ n if it is Σ n and, in addition, the predicate “x codes ∼ 1 ∼ a maximal antichain of P” is also Σ n . ∼ Lightface forms may be defined analogously. Let us recall from [4] that a projective poset P is absolutely-ccc if it is ccc in all inner models of ZFC that contain the parameters of the definition of P. In [4] we proved the equiconsistency of: (1) there exists an inaccessible cardinal; (2) L(R)-two-step absoluteness for strongly-Σ 13 absolutely-ccc posets; and (3) Σ41 ∼ absoluteness for Cohen and Random forcings. We also showed that this is an optimal 1 result since there is a ∆3 , provably in ZFC σ-centered poset P, such that Σ41 absoluteness for P implies that ω1 is a Π1 -Mahlo cardinal in L, where κ is a Π1 Mahlo cardinal iff κ is inaccessible and every club on κ which is Π1 definable in Vκ contains an inaccessible cardinal. Also in [4] we showed the equiconsistency of: (1) there exists a Σ ω -Mahlo cardinal; (2) L(R)-two-step absoluteness for projective ∼ absolutely-ccc posets; and (3) Σ41 -absoluteness for projective absolutely-ccc posets (where a Σ ω -Mahlo cardinal is an inaccessible cardinal κ such that every club subset ∼ of κ which is first-order definable in Vκ , with parameters, contains an inaccessible cardinal). In the present paper we study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every Σ 13 strongly-proper forcing notion (see Definition 5 below) preserves the property of ∼ L(R) being a Solovay model. This yields that the consistency strength of L(R)-twostep absoluteness under Σ 13 strongly-proper forcing notions is that of the existence ∼ of an inaccessible cardinal. Further, we show that every projective strongly-proper forcing notion preserves the property of L(R) being a Σ ω -Mahlo Solovay model. ∼ Hence, L(R)-two-step absoluteness under projective strongly-proper forcing notions is consistent relative to the existence of a Σ ω -Mahlo cardinal. After briefly looking at ∼ the general ccc case and observing that a well-known argument of Kunen, together with a result of Harrington and Shelah shows that the consistency strength of L(R)absoluteness under all Knaster’s (and all ccc) forcing extensions is exactly that of a weakly-compact cardinal, we show that the absoluteness of L(R) under σ-linked forcing notions is equiconsistent with the existence of a Mahlo cardinal. 2. The inaccessible case In this section we prove that the property of being a Solovay model is preserved under any Σ 13 strongly-proper forcing notion. ∼ Recall that a partial order P is proper if for some large-enough regular cardinal λ (e.g., λ > 2|P| ), for a club set of countable elementary substructure N of H(λ) with P ∈ N , and for every p ∈ P ∩ N , there is q ≤ p which is (N, P)-generic, i.e., whenever A ⊆ P is a maximal antichain of P and A ∈ N , then A ∩ N is predense below q. Also, recall that a simple P-name for a real, i.e., a function from ω into ω, is a set of the form {hpn,α , m ˘ n,α i : n < ω, α < κn }, where the m ˘ n,α are standard P-terms for integers and for each n < ω, the set {pn,α : α < κn } is a maximal antichain of P.
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Lemma 3. Suppose that P is a proper forcing notion whose conditions are real numbers. Then for every P-name τ and every condition p ∈ P such that p °P τ ⊆ ω, there is a P-name σ and a condition q ≤ p such that |T C(σ)| = ℵ0 and q ° τ = σ. Proof. We may assume that the elements of τ are of the form hp, ni, where p ∈ P and n ∈ ω. Let N ≺ H(λ), where λ is a large-enough regular cardinal, N countable, and τ, p, P ∈ N . Let q ≤ p be (N, P)-generic. Let σ = τ ∩ N . Then, T C(σ) is countable and q °P σ = τ t u We shall need a stronger notion of properness for projective posets. An even stronger notion appears in M. Goldstern’s Ph. D. thesis [11], in the context of Suslin (Σ 11 ) forcing. ∼ Definition 5. A projective poset P is strongly-proper if for every countable transitive model N of a fragment of ZFC with the parameters of the definition of P in N N N and such that (PN , ≤N P , ⊥P ) ⊆ (P, ≤P , ⊥P ), and for every p ∈ P , there is q ≤ p which is (N, P)-generic, i.e., if N |= “A is a maximal antichain of P”, then A ∩ N is predense below q. Notice that if P is a projective poset, N ¹ H(λ), and the parameters of the ¯ , P)-generic, definition of P are in N , then a condition q is (N, P)-generic iff it is (N ¯ is the transitive collapse of N . Thus, a projective strongly-proper poset is where N proper. All known examples of Suslin proper posets are strongly-proper (see [11]), and it is still an open question from [11] whether every Suslin proper poset is stronglyproper. It follows from the next theorem that the example of a ∆13 provably in ZFC σ-centered poset from [4] is not strongly-proper. Notation 1 For κ a cardinal, if α ≤ κ, we shall write Collα instead of the more cumbersome Coll(ω, < α). If C is a Collκ -generic filter over V , let Cα = C ∩ Collα . If P is a projective poset in V [C], and α < κ, let Pα = PV [Cα ] . We shall need the following fact about the Levy collapse of an inaccessible cardinal, which is an easy consequence of Shoenfield’s Absoluteness Theorem and the fact that Collκ is κ-cc. Fact 1 Suppose that κ is an inaccessible cardinal, ∃xϕ(v0 , ..., vn , x) is a Σ31 formula, where ϕ(v0 , ..., vn , x) is Π21 , and b˙ 0 , ..., b˙ n are simple Collκ -names for reals. Then, for all p ∈ Collκ , p °Collκ ∃xϕ(b˙ 0 , ..., b˙ n , x) iff there exists α < κ and a simple Collα -name for a real c˙ such that for every β, α ≤ β < κ, p °Collβ ϕ(b˙ 0 , ..., b˙ n , c). ˙ Theorem 1. Suppose that L(R)M is a Solovay model over V and P is a Σ 13 strongly∼ proper poset in M . Then the L(R) of any P-generic extension of M is also a Solovay model over V . Proof. Let κ = ω1M . Let P be a Σ 13 strongly-proper poset in M . Force over M with ∼ Woodin’s partial ordering W (Lemma 1) to obtain a Coll(ω, < κ)-generic C over V so that RV [C] = RM . Let r˙ be a simple P-name for a real, r˙ ∈ M . By Lemma 3 we may assume r˙ ∈ V [C]. Suppose that G is P-generic over M . Notice that, since M and V [C] have the same reals, P = PV [C] . Thus, G is generic over V [C].
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V [r[G]] ˙
is countable, it will be enough to show that So, to show that, in M [G], ω1 V [r[G]] ˙ it is countable in V [C][G], for the fact that ω1 is countable is witnessed by a real and, by Lemma 3, P-names for reals are essentially reals. Similarly, for r˙ and G as before, to show that in M [G], r[G] ˙ is small-generic over V , it will be enough to show that this is the case in V [C][G]. Working in V [C], suppose r˙ is a simple P-term, and suppose p ∈ P forces that r˙ : ω → ω. Note that since M and V [C] have the same reals, P is strongly-proper in V [C], hence proper. So, by Lemma 3, let q ∈ P, q ≤ p, and let s˙ be such that 1. T C(s) ˙ is countable. 2. q °P s˙ = r˙ Let α < κ be such that s˙ is a Pα -term in V [Cα ] (use Fact 1). We may also require that p and q are in V [Cα ] and that V [Cα ] |= “p, q ∈ P”. Let β be such that α < β < κ and N =: Vβ [Cα ] is a sufficiently elementary substructure of V [Cα ] which contains all subsets of Pα . Let q 0 ≤ q be (N, P)-generic. Suppose G is P-generic over V [C] with q 0 ∈ G. We claim that Gα := G ∩ Pα is Pα -generic over V [Cα ]. First notice that since P is Σ 13 , by Shoenfield’s absoluteness Gα is a filter. So, ∼ it is enough to see that Gα intersects all A ∈ N such that N |= “A is a maximal antichain of P”. But for any such A, A∩N is predense below q 0 . Hence, G∩A∩N 6= ∅. Therefore, Gα ∩ A 6= ∅. Moreover, since q ∈ Gα , V [C][G] |= r[C ˙ ∗ G] = s[C ˙ α ∗ Gα ]. We have found a partial ordering in V , namely Collα ∗ P˙ α , which is countable in V [C][G], and a Collα ∗ P˙ α -generic Cα ∗ Gα such that r[C ˙ ∗ G] ∈ V [Cα ∗ Gα ]. It follows that every real in V [C][G] is small-generic over V . V [C ∗G ] V [r[G]] ˙ t ≤ ω1 α α < κ. u Finally, for r˙ and α as above, it is clear that ω1 Corollary 1. The following are equiconsistent (modulo ZF C): 1. There exists an inaccessible cardinal. 2. L(R)-two-step absoluteness under Σ 13 strongly-proper forcing. ∼ 3. Σ 14 -absoluteness under Cohen and Random forcing. ∼ Proof. (1) implies (2) follows from Theorem 1. (2) implies (3) follows from the wellknown fact that Cohen and Random forcing are Borel. (3) implies (1) is a result of J. Bagaria and W. H. Woodin (see [1] or [6]). u t 3. The general projective case for strongly-proper forcing notions Let us recall the following definitions from [4]. Definition 6. Let κ be a cardinal. C ⊆ κ is a Π n -closed and unbounded subset ∼ of κ, a Π n -club for short, iff C is a club of κ which is definable in Vκ with a ∼ Πn formula with parameters. A Σ ω -club is a club of κ that is definable in Vκ with ∼ parameters. S ⊆ κ is a Π n -stationary subset of κ iff for all Π n -club C in κ, S ∩ C 6= ∅. ∼ ∼ Similarly, we can define the Σ ω -stationary subsets of κ. ∼ Definition 7. κ is a Π n -Mahlo cardinal iff κ is an inaccessible cardinal and the ∼ set of all inaccessible cardinals below κ is Π n -stationary. ∼ κ is Σ ω -Mahlo if it is Π n -Mahlo for every n < ω. ∼ ∼
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Joan Bagaria, Roger Bosch
Definition 8. L(R)M is a Π n -Mahlo (Σ ω -Mahlo) Solovay model over V ⊆ M iff ∼ ∼ M satisfies: 1. For every x ∈ R, ω1 is a Π n -Mahlo (Σ ω -Mahlo) cardinal in V [x] and ∼ ∼ 2. Every x ∈ R is small-generic over V . We also need the following Lemma from [4]: Lemma 4. Let κ be a cardinal and for every n ≥ 1 let In = {λ < κ : λ is inaccessible and Vλ 4n Vκ }. Then κ is Π n -Mahlo iff κ is inaccessible and In is a Π n -stationary ∼ ∼ subset of κ. Notice that since every Π n -Mahlo (Σ ω -Mahlo) cardinal is inaccessible, Lemma ∼ ∼ 1 also holds for Π n -Mahlo (Σ ω -Mahlo) Solovay models. ∼ ∼ Π n -Mahlo cardinals will allow us to generalize Theorem 1 to all projective proper ∼ posets. For the proof we will need the following generalization of Fact 1 above: 1 formula, Fact 2 Suppose that κ is a Π n -Mahlo cardinal, ∃xϕ(v0 , ..., vn , x) is a Σn+2 ∼ 1 ˙ ˙ where ϕ(v0 , ..., vn , x) is Πn+1 , and b0 , ..., bn are simple Collκ -names for reals. Then, for all p ∈ Collκ , if p °Collκ “∃xϕ(b˙ 0 , ..., b˙ n , x)”
then there exists α < κ and a simple Collα -name for a real c˙ such that for all λ ∈ In greater than α, p °Coll “ϕ(b˙ 0 , ..., b˙ n , c)”. ˙ λ
This follows immediately from the following two facts about the Levy collapse, which are proved in [4]: Fact 3 Let n ≥ 2 and let θ(x) be a Σn1 (Πn1 ) formula. Suppose that κ is an inaccessible cardinal. Then the relation R(p, τ ) iff τ is a simple Collκ -term for a real and p °Collκ θ(τ ), is definable in Vκ . Moreover, 1. If θ(x) is a Σn1 formula, then R is Σn−1 in Vκ . 2. If θ(x) is a Πn1 formula, then R is Πn−1 in Vκ . Fact 4 Suppose that κ is an inaccessible cardinal and θ(x) is a Σn1 (Πn1 ) formula. Then, for every p ∈ Collκ and every simple Collκ -name for a real τ , p °Collκ θ (τ ) iff Vκ |= p °Collκ θ (τ ) . Theorem 2. Let n ≥ 2. Suppose L(R)M is a Π n -Solovay model over V and P is a ∼ Σ 1n+2 strongly-proper poset in M . Then the L(R) of any P-generic extension of M ∼ is also a Π n -Solovay model over V . ∼ Proof. The proof is as in 1, but now the formulas ϕ≤ (x, y) and ϕ⊥ (x, y) that define 1 the ordering and the incompatibility relations on P are Σn+2 . We may assume the real parameters of these formulas are in V . By Fact 3, for every p ∈ Collκ and every simple Collκ -names for reals τ, σ, “p °Collκ ϕ≤ (τ, σ) ” and “p °Collκ ϕ⊥ (τ, σ) ” are expressible in Vκ with Σn+1 formulas with parameters p, τ , σ and the real parameters of ϕ≤ and ϕ⊥ . By Lemma 4, we can find a Π n -stationary sequence hλξ : ξ < κi such that for ∼ every ξ < κ, λξ is an inaccessible cardinal and Vλξ 4n Vκ . Let p ∈ Collκ and let σ, τ be simple Collκ -names for reals. There is a least ξ < κ such that p ∈ Collλξ and σ and τ are simple Collλξ -names. Then, for every ζ < κ with ξ ≤ ζ, by facts 4 and 3, if p °Collλζ ϕ≤ (σ, τ ), then p °Collκ ϕ≤ (σ, τ ), and if p °Collλζ ϕ⊥ (σ, τ ), then p °Collκ ϕ⊥ (σ, τ ). Thus, we have that for every ξ < κ, Pλξ ⊆ P, ≤Pλξ ⊆≤P , and ⊥Pλξ ⊆⊥P . The rest of the proof proceeds as in the proof of Theorem 1 by taking a sufficiently large λξ instead of α (see the proof of Theorem 1). u t
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Corollary 2. 1. For every n ≥ 2, Con(ZF C + ∃κ (κ is a Π n -Mahlo cardinal)) im∼ plies Con(ZF C + L (R)-two-step absoluteness for Σ 1n+2 strongly-proper posets). ∼ 2. Con(ZF C + ∃κ (κ is a Σ ω -Mahlo cardinal)) implies Con(ZF C + L (R)-two-step ∼ absoluteness for projective strongly-proper posets). It is still an open question whether the consistency strength of L (R)-two-step absoluteness for projective strongly-proper posets is exactly that of the existence of a Σ ω -Mahlo cardinal. ∼ We will next consider the absoluteness of L(R) under arbitrary projective ccc forcing notions. 4. The general ccc case For the general ccc case a weakly-compact cardinal is required. This follows from some well-known results of Kunen and Harrington-Shelah. M
Definition 9. L (R) satisfies:
is a weakly-compact Solovay model over V ⊆ M iff M
1. For every x ∈ R, ω1 is a weakly-compact cardinal in V [x] and 2. For every x ∈ R, V [x] is a generic extension of V by some countable poset. The following theorem is essentially due to Kunen (see [9]): Theorem 3. Suppose κ is a weakly-compact cardinal in V , C is Collκ -generic over V , and P is a ccc poset in V [C]. Then the L(R) of any P-generic extension of V [C] is a weakly-compact Solovay model over V . Definition 10. A poset P is Knaster iff for all uncountable subset X of P there exists an uncountable Y ⊆ X of pairwise compatible conditions. Theorem 4. The following are equiconsistent (modulo ZF C): 1. There exists a weakly-compact cardinal. 2. L(R)-two-step-absoluteness for ccc forcing extensions. 3. L(R)-two-step-absoluteness for Knaster forcing extensions. 4. Σ41 -absoluteness for Knaster forcing extensions. Proof. (1) implies (2) follows from Theorem 3. (2) implies (3) and (3) implies (4) are trivial. (4) implies (1): Let V be a model of ZF C that is Σ41 -absolute under Knaster forcing notions. Suppose that ω1 is not weakly-compact in L. Then, by an argument of J.H. Silver (see [8]), we can find an Aronszajn tree T ∈ L on ω1 such that for every model M of ZF C, if M ² “T has a branch of length ω1 ”, then M ² cf (ω1 ) = ω. Without loss of generality, we may assume that T has infinitely many nodes of height 0. We now use the Knaster poset P(T, (dα )α
Joan Bagaria1 · Roger Bosch2
Proper forcing extensions and Solovay models c Springer-Verlag 2003 Received: date / Revised version: date – ° Abstract. We study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every Σ 13 strongly-proper forcing notion ∼ preserves this property. This yields that the consistency strength of the absoluteness of 1 L(R) under Σ 3 strongly-proper forcing notions is that of the existence of an inaccessible ∼ cardinal. Further, the absoluteness of L(R) under projective strongly-proper forcing notions is consistent relative to the existence of a Σ ω -Mahlo cardinal. We also show that the ∼ consistency strength of the absoluteness of L(R) under forcing extensions with σ-linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.
1. Introduction A Solovay model is, essentially, the model discovered by Solovay in [12], in which every set of reals is Lebesgue measurable, has the property of Baire, has the perfect set property, etc. That is, if κ is an inaccessible cardinal in some model V , then the L(R) of a model M resulting from collapsing κ to ω1 over V using the Levy collapse Coll(ω, < κ) is a Solovay model over V . In this situation, M has the following properties: 1. For every x ∈ R, ω1 is an inaccessible cardinal in V [x]. 2. Every x ∈ R is small-generic over V . That is, there is a forcing notion P in V , which is countable in M , and there is, in M , a P-generic filter g over V such that x ∈ V [g]. Woodin has showed (see [4]) that the converse is essentially true. Namely, Lemma 1. Suppose that V ⊆ M are models of (a fragment of ) ZFC and M satisfies (1) and (2) above. Then there is a forcing notion W in M which does not add new reals and creates a Coll (ω, < ω1 )-generic filter C over V such that M and V [C] have the same reals. 1: Instituci´ o Catalana de Recerca i Estudis Avan¸cats (ICREA) and Departament de L` ogica, Hist` oria i Filosofia de la Ci`encia. Universitat de Barcelona. Baldiri Reixac, s/n. 08028 Barcelona, Catalonia (Spain) e-mail: [email protected] 2: Departamento de Filosof´ıa. Universidad de Oviedo. Teniente Alfonso Mart´ınez, s/n. 33011 Oviedo, Asturias (Spain) e-mail: [email protected] Research partially supported by the research projects BFM2002-03236 of the Spanish Ministry of Science and Technology, and 2002SGR 00126 of the Generalitat de Catalunya. The second author was also partially supported by the research project GE01/HUM10, Grupos de excelencia, Principado de Asturias. Key words or phrases: Solovay models – generic absoluteness – strongly-proper forcing – projective forcing – consistency strength – definably-Mahlo cardinals – Mahlo cardinals – weakly-compact cardinals.
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The conditions of W are filters which are generic over V for some initial segment of the Levy-collapse of ω1M . Thus, W forces that L(R)M is a Solovay model over V . So, in view of this we define: Definition 1. L(R)M is a Solovay model over V ⊆ M iff M satisfies: 1. For every x ∈ R, ω1 is an inaccessible cardinal in V [x] and 2. Every x ∈ R is small-generic over V . Our interest of the preservation of the property of being a Solovay model under forcing extensions that do not collapse ω1 lies mainly in the fact that it implies a strong form of generic absoluteness, which in turn has strong consequences for the theory of the reals (see the forthcomming [2]). By generic absoluteness we mean the following: Definition 2. Let V be a model of ZF . Let P ∈ V be a forcing notion and let ϕ be a formula (possibly with parameters in V ). V is ϕ-absolute for P iff V |= ϕ iff V P |= ϕ. If Σ is a set of formulas, V is Σ-absolute for P iff for every ϕ ∈ Σ, V is ϕ-absolute for P. Given a class of posets Γ , V is Σ-absolute for Γ iff for every P ∈ Γ , V is Σ-absolute for P in V . V is L(R)-absolute for P iff there exists an elementary embedding j : L(R)V → L(R)V
P
that fixes all the ordinals (and, of course, all the reals). For Γ a class of posets, V is L(R)-absolute for Γ if it is L(R)-absolute for every P in Γ . The main fact connecting Solovay models and generic absoluteness is given by the following lemma (see [4]). Lemma 2. Suppose that L(R)M and L(R)N are Solovay models over V such that RM ⊆ RN and ω1M = ω1N . Then there exists an elementary embedding j : L(R)M → L(R)N which fixes all the ordinals. An even stronger form of generic absoluteness is the following (see [4]): ˙ a P-name for a forcing notion. Definition 3. Let P be a forcing notion in V , and Q ˙ V is two-step ϕ-absolute for P and Q , if ˙
V P |= ϕ iff V P∗Q |= ϕ. (Note that ϕ may have parameters in V P , not just in V .) For Σ a set of formulas, ˙ in the obvious way. For Γ a class we define V is two-step Σ-absolute for P and Q ˙ of posets, V is two-step Σ-absolute for Γ iff for every P ∈ Γ and every P-name Q ˙ for a forcing notion in Γ , Σ is absolute for P and Q. ˙ iff there exists an elementary embedding V is L(R)-two-step absolute for P and Q P
j : L(R)V → L(R)V
˙ P∗Q
that fixes all the ordinals. For Γ a class of posets, V is L(R)-two-step absolute for Γ iff for every P ∈ Γ ˙ for a forcing notion in Γ , V is L(R)-two-step absolute for P and every P-name Q ˙ and Q.
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This paper continues the work started in [4] where we proved several equiconsistency results on generic absoluteness for projective ccc posets. More precisely, Definition 4. A Σ 1n poset (n ≥ 1) is a triple P = hP, ≤P , ⊥P i, where ≤P is a Σ 1n ∼ ∼ subset of ω ω × ω ω , P = field(≤P ), hP, ≤P i is a partial order, and ⊥P is a Σ 1n subset ∼ of ω ω × ω ω contained in P × P such that for every x, y ∈ P , x ⊥P y iff for no z ∈ P we have that z ≤P x and z ≤P y; i.e., iff x, y are incompatible. Similarly, we define Π 1n posets by substituting Π 1n for Σ 1n in the above definition. A ∆1n poset is a poset ∼ ∼ ∼ ∼ that is both Σ 1n and Π 1n . Finally, P is a projective poset iff P is Σ 1n , for some n. ∼ ∼ ∼ 1 1 A ccc poset P is strongly-Σ n if it is Σ n and, in addition, the predicate “x codes ∼ 1 ∼ a maximal antichain of P” is also Σ n . ∼ Lightface forms may be defined analogously. Let us recall from [4] that a projective poset P is absolutely-ccc if it is ccc in all inner models of ZFC that contain the parameters of the definition of P. In [4] we proved the equiconsistency of: (1) there exists an inaccessible cardinal; (2) L(R)-two-step absoluteness for strongly-Σ 13 absolutely-ccc posets; and (3) Σ41 ∼ absoluteness for Cohen and Random forcings. We also showed that this is an optimal 1 result since there is a ∆3 , provably in ZFC σ-centered poset P, such that Σ41 absoluteness for P implies that ω1 is a Π1 -Mahlo cardinal in L, where κ is a Π1 Mahlo cardinal iff κ is inaccessible and every club on κ which is Π1 definable in Vκ contains an inaccessible cardinal. Also in [4] we showed the equiconsistency of: (1) there exists a Σ ω -Mahlo cardinal; (2) L(R)-two-step absoluteness for projective ∼ absolutely-ccc posets; and (3) Σ41 -absoluteness for projective absolutely-ccc posets (where a Σ ω -Mahlo cardinal is an inaccessible cardinal κ such that every club subset ∼ of κ which is first-order definable in Vκ , with parameters, contains an inaccessible cardinal). In the present paper we study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every Σ 13 strongly-proper forcing notion (see Definition 5 below) preserves the property of ∼ L(R) being a Solovay model. This yields that the consistency strength of L(R)-twostep absoluteness under Σ 13 strongly-proper forcing notions is that of the existence ∼ of an inaccessible cardinal. Further, we show that every projective strongly-proper forcing notion preserves the property of L(R) being a Σ ω -Mahlo Solovay model. ∼ Hence, L(R)-two-step absoluteness under projective strongly-proper forcing notions is consistent relative to the existence of a Σ ω -Mahlo cardinal. After briefly looking at ∼ the general ccc case and observing that a well-known argument of Kunen, together with a result of Harrington and Shelah shows that the consistency strength of L(R)absoluteness under all Knaster’s (and all ccc) forcing extensions is exactly that of a weakly-compact cardinal, we show that the absoluteness of L(R) under σ-linked forcing notions is equiconsistent with the existence of a Mahlo cardinal. 2. The inaccessible case In this section we prove that the property of being a Solovay model is preserved under any Σ 13 strongly-proper forcing notion. ∼ Recall that a partial order P is proper if for some large-enough regular cardinal λ (e.g., λ > 2|P| ), for a club set of countable elementary substructure N of H(λ) with P ∈ N , and for every p ∈ P ∩ N , there is q ≤ p which is (N, P)-generic, i.e., whenever A ⊆ P is a maximal antichain of P and A ∈ N , then A ∩ N is predense below q. Also, recall that a simple P-name for a real, i.e., a function from ω into ω, is a set of the form {hpn,α , m ˘ n,α i : n < ω, α < κn }, where the m ˘ n,α are standard P-terms for integers and for each n < ω, the set {pn,α : α < κn } is a maximal antichain of P.
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Joan Bagaria, Roger Bosch
Lemma 3. Suppose that P is a proper forcing notion whose conditions are real numbers. Then for every P-name τ and every condition p ∈ P such that p °P τ ⊆ ω, there is a P-name σ and a condition q ≤ p such that |T C(σ)| = ℵ0 and q ° τ = σ. Proof. We may assume that the elements of τ are of the form hp, ni, where p ∈ P and n ∈ ω. Let N ≺ H(λ), where λ is a large-enough regular cardinal, N countable, and τ, p, P ∈ N . Let q ≤ p be (N, P)-generic. Let σ = τ ∩ N . Then, T C(σ) is countable and q °P σ = τ t u We shall need a stronger notion of properness for projective posets. An even stronger notion appears in M. Goldstern’s Ph. D. thesis [11], in the context of Suslin (Σ 11 ) forcing. ∼ Definition 5. A projective poset P is strongly-proper if for every countable transitive model N of a fragment of ZFC with the parameters of the definition of P in N N N and such that (PN , ≤N P , ⊥P ) ⊆ (P, ≤P , ⊥P ), and for every p ∈ P , there is q ≤ p which is (N, P)-generic, i.e., if N |= “A is a maximal antichain of P”, then A ∩ N is predense below q. Notice that if P is a projective poset, N ¹ H(λ), and the parameters of the ¯ , P)-generic, definition of P are in N , then a condition q is (N, P)-generic iff it is (N ¯ is the transitive collapse of N . Thus, a projective strongly-proper poset is where N proper. All known examples of Suslin proper posets are strongly-proper (see [11]), and it is still an open question from [11] whether every Suslin proper poset is stronglyproper. It follows from the next theorem that the example of a ∆13 provably in ZFC σ-centered poset from [4] is not strongly-proper. Notation 1 For κ a cardinal, if α ≤ κ, we shall write Collα instead of the more cumbersome Coll(ω, < α). If C is a Collκ -generic filter over V , let Cα = C ∩ Collα . If P is a projective poset in V [C], and α < κ, let Pα = PV [Cα ] . We shall need the following fact about the Levy collapse of an inaccessible cardinal, which is an easy consequence of Shoenfield’s Absoluteness Theorem and the fact that Collκ is κ-cc. Fact 1 Suppose that κ is an inaccessible cardinal, ∃xϕ(v0 , ..., vn , x) is a Σ31 formula, where ϕ(v0 , ..., vn , x) is Π21 , and b˙ 0 , ..., b˙ n are simple Collκ -names for reals. Then, for all p ∈ Collκ , p °Collκ ∃xϕ(b˙ 0 , ..., b˙ n , x) iff there exists α < κ and a simple Collα -name for a real c˙ such that for every β, α ≤ β < κ, p °Collβ ϕ(b˙ 0 , ..., b˙ n , c). ˙ Theorem 1. Suppose that L(R)M is a Solovay model over V and P is a Σ 13 strongly∼ proper poset in M . Then the L(R) of any P-generic extension of M is also a Solovay model over V . Proof. Let κ = ω1M . Let P be a Σ 13 strongly-proper poset in M . Force over M with ∼ Woodin’s partial ordering W (Lemma 1) to obtain a Coll(ω, < κ)-generic C over V so that RV [C] = RM . Let r˙ be a simple P-name for a real, r˙ ∈ M . By Lemma 3 we may assume r˙ ∈ V [C]. Suppose that G is P-generic over M . Notice that, since M and V [C] have the same reals, P = PV [C] . Thus, G is generic over V [C].
Proper forcing extensions and Solovay models
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V [r[G]] ˙
is countable, it will be enough to show that So, to show that, in M [G], ω1 V [r[G]] ˙ it is countable in V [C][G], for the fact that ω1 is countable is witnessed by a real and, by Lemma 3, P-names for reals are essentially reals. Similarly, for r˙ and G as before, to show that in M [G], r[G] ˙ is small-generic over V , it will be enough to show that this is the case in V [C][G]. Working in V [C], suppose r˙ is a simple P-term, and suppose p ∈ P forces that r˙ : ω → ω. Note that since M and V [C] have the same reals, P is strongly-proper in V [C], hence proper. So, by Lemma 3, let q ∈ P, q ≤ p, and let s˙ be such that 1. T C(s) ˙ is countable. 2. q °P s˙ = r˙ Let α < κ be such that s˙ is a Pα -term in V [Cα ] (use Fact 1). We may also require that p and q are in V [Cα ] and that V [Cα ] |= “p, q ∈ P”. Let β be such that α < β < κ and N =: Vβ [Cα ] is a sufficiently elementary substructure of V [Cα ] which contains all subsets of Pα . Let q 0 ≤ q be (N, P)-generic. Suppose G is P-generic over V [C] with q 0 ∈ G. We claim that Gα := G ∩ Pα is Pα -generic over V [Cα ]. First notice that since P is Σ 13 , by Shoenfield’s absoluteness Gα is a filter. So, ∼ it is enough to see that Gα intersects all A ∈ N such that N |= “A is a maximal antichain of P”. But for any such A, A∩N is predense below q 0 . Hence, G∩A∩N 6= ∅. Therefore, Gα ∩ A 6= ∅. Moreover, since q ∈ Gα , V [C][G] |= r[C ˙ ∗ G] = s[C ˙ α ∗ Gα ]. We have found a partial ordering in V , namely Collα ∗ P˙ α , which is countable in V [C][G], and a Collα ∗ P˙ α -generic Cα ∗ Gα such that r[C ˙ ∗ G] ∈ V [Cα ∗ Gα ]. It follows that every real in V [C][G] is small-generic over V . V [C ∗G ] V [r[G]] ˙ t ≤ ω1 α α < κ. u Finally, for r˙ and α as above, it is clear that ω1 Corollary 1. The following are equiconsistent (modulo ZF C): 1. There exists an inaccessible cardinal. 2. L(R)-two-step absoluteness under Σ 13 strongly-proper forcing. ∼ 3. Σ 14 -absoluteness under Cohen and Random forcing. ∼ Proof. (1) implies (2) follows from Theorem 1. (2) implies (3) follows from the wellknown fact that Cohen and Random forcing are Borel. (3) implies (1) is a result of J. Bagaria and W. H. Woodin (see [1] or [6]). u t 3. The general projective case for strongly-proper forcing notions Let us recall the following definitions from [4]. Definition 6. Let κ be a cardinal. C ⊆ κ is a Π n -closed and unbounded subset ∼ of κ, a Π n -club for short, iff C is a club of κ which is definable in Vκ with a ∼ Πn formula with parameters. A Σ ω -club is a club of κ that is definable in Vκ with ∼ parameters. S ⊆ κ is a Π n -stationary subset of κ iff for all Π n -club C in κ, S ∩ C 6= ∅. ∼ ∼ Similarly, we can define the Σ ω -stationary subsets of κ. ∼ Definition 7. κ is a Π n -Mahlo cardinal iff κ is an inaccessible cardinal and the ∼ set of all inaccessible cardinals below κ is Π n -stationary. ∼ κ is Σ ω -Mahlo if it is Π n -Mahlo for every n < ω. ∼ ∼
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Joan Bagaria, Roger Bosch
Definition 8. L(R)M is a Π n -Mahlo (Σ ω -Mahlo) Solovay model over V ⊆ M iff ∼ ∼ M satisfies: 1. For every x ∈ R, ω1 is a Π n -Mahlo (Σ ω -Mahlo) cardinal in V [x] and ∼ ∼ 2. Every x ∈ R is small-generic over V . We also need the following Lemma from [4]: Lemma 4. Let κ be a cardinal and for every n ≥ 1 let In = {λ < κ : λ is inaccessible and Vλ 4n Vκ }. Then κ is Π n -Mahlo iff κ is inaccessible and In is a Π n -stationary ∼ ∼ subset of κ. Notice that since every Π n -Mahlo (Σ ω -Mahlo) cardinal is inaccessible, Lemma ∼ ∼ 1 also holds for Π n -Mahlo (Σ ω -Mahlo) Solovay models. ∼ ∼ Π n -Mahlo cardinals will allow us to generalize Theorem 1 to all projective proper ∼ posets. For the proof we will need the following generalization of Fact 1 above: 1 formula, Fact 2 Suppose that κ is a Π n -Mahlo cardinal, ∃xϕ(v0 , ..., vn , x) is a Σn+2 ∼ 1 ˙ ˙ where ϕ(v0 , ..., vn , x) is Πn+1 , and b0 , ..., bn are simple Collκ -names for reals. Then, for all p ∈ Collκ , if p °Collκ “∃xϕ(b˙ 0 , ..., b˙ n , x)”
then there exists α < κ and a simple Collα -name for a real c˙ such that for all λ ∈ In greater than α, p °Coll “ϕ(b˙ 0 , ..., b˙ n , c)”. ˙ λ
This follows immediately from the following two facts about the Levy collapse, which are proved in [4]: Fact 3 Let n ≥ 2 and let θ(x) be a Σn1 (Πn1 ) formula. Suppose that κ is an inaccessible cardinal. Then the relation R(p, τ ) iff τ is a simple Collκ -term for a real and p °Collκ θ(τ ), is definable in Vκ . Moreover, 1. If θ(x) is a Σn1 formula, then R is Σn−1 in Vκ . 2. If θ(x) is a Πn1 formula, then R is Πn−1 in Vκ . Fact 4 Suppose that κ is an inaccessible cardinal and θ(x) is a Σn1 (Πn1 ) formula. Then, for every p ∈ Collκ and every simple Collκ -name for a real τ , p °Collκ θ (τ ) iff Vκ |= p °Collκ θ (τ ) . Theorem 2. Let n ≥ 2. Suppose L(R)M is a Π n -Solovay model over V and P is a ∼ Σ 1n+2 strongly-proper poset in M . Then the L(R) of any P-generic extension of M ∼ is also a Π n -Solovay model over V . ∼ Proof. The proof is as in 1, but now the formulas ϕ≤ (x, y) and ϕ⊥ (x, y) that define 1 the ordering and the incompatibility relations on P are Σn+2 . We may assume the real parameters of these formulas are in V . By Fact 3, for every p ∈ Collκ and every simple Collκ -names for reals τ, σ, “p °Collκ ϕ≤ (τ, σ) ” and “p °Collκ ϕ⊥ (τ, σ) ” are expressible in Vκ with Σn+1 formulas with parameters p, τ , σ and the real parameters of ϕ≤ and ϕ⊥ . By Lemma 4, we can find a Π n -stationary sequence hλξ : ξ < κi such that for ∼ every ξ < κ, λξ is an inaccessible cardinal and Vλξ 4n Vκ . Let p ∈ Collκ and let σ, τ be simple Collκ -names for reals. There is a least ξ < κ such that p ∈ Collλξ and σ and τ are simple Collλξ -names. Then, for every ζ < κ with ξ ≤ ζ, by facts 4 and 3, if p °Collλζ ϕ≤ (σ, τ ), then p °Collκ ϕ≤ (σ, τ ), and if p °Collλζ ϕ⊥ (σ, τ ), then p °Collκ ϕ⊥ (σ, τ ). Thus, we have that for every ξ < κ, Pλξ ⊆ P, ≤Pλξ ⊆≤P , and ⊥Pλξ ⊆⊥P . The rest of the proof proceeds as in the proof of Theorem 1 by taking a sufficiently large λξ instead of α (see the proof of Theorem 1). u t
Proper forcing extensions and Solovay models
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Corollary 2. 1. For every n ≥ 2, Con(ZF C + ∃κ (κ is a Π n -Mahlo cardinal)) im∼ plies Con(ZF C + L (R)-two-step absoluteness for Σ 1n+2 strongly-proper posets). ∼ 2. Con(ZF C + ∃κ (κ is a Σ ω -Mahlo cardinal)) implies Con(ZF C + L (R)-two-step ∼ absoluteness for projective strongly-proper posets). It is still an open question whether the consistency strength of L (R)-two-step absoluteness for projective strongly-proper posets is exactly that of the existence of a Σ ω -Mahlo cardinal. ∼ We will next consider the absoluteness of L(R) under arbitrary projective ccc forcing notions. 4. The general ccc case For the general ccc case a weakly-compact cardinal is required. This follows from some well-known results of Kunen and Harrington-Shelah. M
Definition 9. L (R) satisfies:
is a weakly-compact Solovay model over V ⊆ M iff M
1. For every x ∈ R, ω1 is a weakly-compact cardinal in V [x] and 2. For every x ∈ R, V [x] is a generic extension of V by some countable poset. The following theorem is essentially due to Kunen (see [9]): Theorem 3. Suppose κ is a weakly-compact cardinal in V , C is Collκ -generic over V , and P is a ccc poset in V [C]. Then the L(R) of any P-generic extension of V [C] is a weakly-compact Solovay model over V . Definition 10. A poset P is Knaster iff for all uncountable subset X of P there exists an uncountable Y ⊆ X of pairwise compatible conditions. Theorem 4. The following are equiconsistent (modulo ZF C): 1. There exists a weakly-compact cardinal. 2. L(R)-two-step-absoluteness for ccc forcing extensions. 3. L(R)-two-step-absoluteness for Knaster forcing extensions. 4. Σ41 -absoluteness for Knaster forcing extensions. Proof. (1) implies (2) follows from Theorem 3. (2) implies (3) and (3) implies (4) are trivial. (4) implies (1): Let V be a model of ZF C that is Σ41 -absolute under Knaster forcing notions. Suppose that ω1 is not weakly-compact in L. Then, by an argument of J.H. Silver (see [8]), we can find an Aronszajn tree T ∈ L on ω1 such that for every model M of ZF C, if M ² “T has a branch of length ω1 ”, then M ² cf (ω1 ) = ω. Without loss of generality, we may assume that T has infinitely many nodes of height 0. We now use the Knaster poset P(T, (dα )α
