Axiomatization of generic extensions by homogeneous partial orderings

Transcription

1 Axiomatization of generic extensions by homogeneous partial orderings a talk at Colloquium on Mathematical Logic (Amsterdam Utrecht) May 29, 2008 (Sakaé Fuchino) Chubu Univ., (CRM Barcelona) ( :57 )

2 Axiomatization of generic extensions... *** Main References *** (2/13) [1] J. Brendle and S.F.: Coloring ordinals by reals, Fundamenta Mathematicae, 196, No.2 (2007), [2] S.F.: A generalization of a problem of Fremlin, RIMS Kôkyûroku, No.1595 (2008), [3] S.F., S. Geschke and L. Soukup: On the weak Freese-Nation property of P(ω), Archive for Mathematical Logic, Vol.40 (2001), [4] S.F. and L. Soukup: More set-theory around the weak Freese-Nation property, Fundamenta Matematicae 154 (1997),

3 Axiomatization of generic extensions... *** Axiomatization of Cohen models *** (3/13) A Boolean algebra (B, B ) has the weak Freese-Nation property iff there is a mapping f : B [B] ℵ 0 such that, for all a, b B, when ever a B b there is c f(a) f(b) such that a B c B b. WFN: (P(ω), ) has the weak Freese-Nation property. Theorem 1. (S.F. and L.Soukup [4]) (1) If W is a model of set theory obtained by adding ℵ n many Cohen reals for n = 1, 2, 3,..., to a model V of CH then we have W = WFN. (2) If W is a model of set theory obtained by adding regular cardinal many Cohen reals to a model V of V = L (actually much weaker condition is enough) then we have W = WFN. Fact 2. (S.F., S.Geschke and L.Soukup [3]) WFN implies may/almost all properties of Cohen models. In particular, the values of all of the usual cardinal invariants of the reals are decided to be as in a Cohen model. Conclusion. WFN may be seen as a natural axiomatization of Cohen models.

4 Axiomatization of generic extensions... *** Axiomatization of Cohen models *** (4/13) Note that WFN follows from CH, since any Boolean algebra of cardinality ℵ 1 has the weak Freese-Nation property. If some statement is proved to be ture in a Cohen model, it is often a good question to ask whether this statement follows from WFN. One of the modst recent applications of WFN: Theorem 3. Assume WFN. (M.Elekes, T. Mátrai and L.Soukup, still unpublished) If U is a covering of R by closed sets such that each x R is included in uncountably many elements of U (ℵ 1 -fold covering) then U can be partitioned into uncountably many (pairwise disjoint) subcoverings. It is also known that under MA the negation of the statement of the theorem above can be proved.

5 Axiomatization of generic extensions... *** C s (κ) *** (5/13) Notation: For any sets X, X 0,..., X n 1 let ((X)) n = { x X n : x is injective}, ((X)) <ω = n<ω ((X))n and ((X 0,..., X n 1 )) = { x X 0 X n 1 : x is injective}. For regular cardinal κ ℵ 2, let C s (κ): For any matrix a α,n : α κ, n ω of subsets of ω and T ω> ω, at least one of the following holds: (c0) there is a stationary S κ such that n< t a α n,t(n) for all t T and for all α 0,..., α t 1 ((S)) <ω ; (c1) there exist t T and stationary S 0,..., S t 1 κ such that n< t a α n,t(n) = for all α 0,..., α t 1 ((S 0,..., S t 1 )). Theorem 4. (I. Juhász, L. Soukup and Z. Szentmiklóssy, 1995) C s (κ) for regular κ ℵ 2 holds in a Cohen model. Theorem 5. (S. Shelah, (unpublished?) note for I. Juhász 2002) (A weakening of) WFN implies C s (κ) for all regular κ ℵ 2.

6 Axiomatization of generic extensions... *** Homogeneity Principle *** (6/13) HP(κ): For any f : κ P(ω) and any projective A ((P(ω))) <ω, at least one of the following holds: (h0) there is a stationary S κ such that ((f S)) <ω \ { } A; (h1) there are k ω \ 1 and stationary S 0,..., S k 1 κ such that ((f S 0,..., f S k 1 )) A =. Theorem 6. (J.Brendle and S.F. [1]) HP(κ) implies C s (κ). Theorem 7. (J.Brendle and S.F. [1]) (a) Assume CH and P = Fn(µ, 2) for some cardinal µ. Then P HP(ℵ 2 ) holds. (b) Assume GCH and P = Fn(µ, 2) for some cardinal µ. Then P HP(κ + ) holds for every κ of uncountable cofinality and P HP(λ) for every inaccessible λ. (c) Assume CH and P is a finite support product of copies of a productively c.c.c. poset of cardinality ℵ 1. Then P HP(ℵ 2 ). (d) Assume GCH and P is a finite support product of copies of a productively c.c.c. poset of cardinality ℵ 1. Then P HP(κ + ) holds for every κ of uncountable cofinality and P HP(λ) for every inaccessible λ.

7 Axiomatization of generic extensions... *** Homogeneity Principle *** (7/13) (e) Assume CH and P is a countable support product of copies of a proper poset of cardinality ℵ 1 such that its product is also proper. Then P HP(ℵ 2 ) holds. (f) Assume GCH and P is a countable support product of copies of a proper poset of cardinality ℵ 1 such that its product is also proper. Then P HP(κ + ) holds for every κ of uncountable cofinality and P HP(λ) for every inaccessible λ. Note that countable support products of Sacks or Prikry-Silver forcing are instances of (e) and (f) above. A [ω] ℵ 0 is said to be a κ-lusin gap if A = κ and for any x [ω]ℵ 0 either {a A : a \ x < ℵ 0 } < κ or {a A : a x < ℵ 0 } < κ. K. Kunen proved that there is a κ-lusin gap for an uncountable κ in a random model (i.e. a model obtained by adding κ random reals side-by-side to a ground model of CH). On the other hand, I. Juhász, L. Soukup and Z. Szentmiklóssy proved that there is no κ-lusin gap under C s (κ). This proves that C s (κ) does not hold in such a model.

8 Axiomatization of generic extensions...*** An application of Homogeneity Principle *** (8/13) Let do = sup{cf( X, R X ) : X ω ω, R is a projective binary Theorem 8. HP(κ) implies do κ. If do is attained then HP(κ) implies do < κ. relation and R X 2 well orders X} Proof. Suppose that there is a projective R ( ω ω) 2 and X ω ω such that otp(x, R X 2 ) = κ. Let f : κ ω ω be defined by f(α) = the α th element of X with respect to R X 2 and let A = R k ω\{2} ((ω ω)) k. Then f, A = (h0) and f, A = (h1). Theorem 9. and do = ℵ 2. (I.Juhász and K.Kunen, 2001) It is consistent that C s (ℵ 2 ) holds Corollary 10. HP(ℵ 2 ) is strictly stronger than C s (ℵ 2 ).

9 Axiomatization of generic extensions... *** Fremlin-Miller Covering Property *** (9/13) Question (D. Fremlin 198?) Is it consistent that the continuum is ℵ 3 and (1) for every family F of Borel sets of size ℵ 2, if F has empty intersection then some subfamily of F of size ℵ 1 has empty intersection? By taking complements, (1) is equivalent to: (1 ) for every family F of Borel sets of size ℵ 2, if R = F then for some G F of cardinality ℵ 1, R = G. Theorem 11. (A. Miller, 1989) Starting from a model of CH, if ℵ 3 Cohen reals are added, the covering property above holds in the resulting model.

10 Axiomatization of generic extensions... *** Fremlin-Miller Covering Property *** (10/13) For cardinals κ λ FCP(κ, λ) : For any family F of Borel sets with F < κ such that F = there is F [F] <λ such that F =. Lemma 12. (0) For κ κ λ λ, FCP(κ, λ) implies FCP(κ, λ ). (1) FCP(κ, κ) holds for any cardinal κ. (2) FCP(c +, c) does not hold. (3) FCP(ℵ 2, ℵ 1 ) does not hold. (4) If κ is one of a, b,... etc. then FCP(κ +, κ) does not hold.

11 Axiomatization of generic extensions... *** Fremlin-Miller Covering Property *** (11/13) Lemma 12. (0) For κ κ λ λ, FCP(κ, λ) implies FCP(κ, λ ). (1) FCP(κ, κ) holds for any cardinal κ. (2) FCP(c +, c) does not hold. (3) FCP(ℵ 2, ℵ 1 ) does not hold. (4) If κ is one of a, b,... etc. then FCP(κ +, κ) does not hold. Proof. (0), (1): trivial. (2): Let F = {R \ {a} : a R}. (3): Let f α α<ω1, g β β<ω1 be a Hausdorff gap. For each α < ω 1, let X α = {f ω ω : f α f g α }. X α s are Borel sets. α<ω 1 X α = but α I X α for any I [ω 1 ] ℵ 0. (4): For κ = a, let A be a maximal almost disjoint family [ω] ℵ 0 of cardinality κ. For each a A, let X a = {x P(ω) : x is almost disjoint from a}. X a Borel(P(ω)) for all a A. a A X a = by the maximality of A but a A X a for any A A. The first nontrivial instance of FCP is FCP(ℵ 3, ℵ 2 ) under c ℵ 3. A. Miller s result can be reformulated as V [G] = FCP(ℵ 3, ℵ 2 ) for V = CH and G adding ℵ 3 Cohen reals.

12 Axiomatization of generic extensions... *** Further Generalization *** (12/13) GFCP(κ, λ) : For any projective relation R R 2, and X [R] <κ, if X is unbounded in R, R, there is X 0 [X] <λ such that X 0 is unbounded in R, R. X is unbounded in R, R r R x X (x R r). Proposition 13. GFCP(κ, λ) implies FCP(κ, λ) for any cardinals κ λ. Proof. Assume that GFCP(κ, λ) holds and suppose that X α : α < δ is a sequence of Borel subsets of R for some δ < κ such that α<δ X α =. For α < δ, let c α be a Borel { code of X α and let X = {c α : α < δ}. For any x R, let the Borel set coded by x, if x is a Borel code Bx =, otherwise. Let x R y B y is a non empty subset of B x for x, y R. The relation R is easily seen to be Π 1 1. Clearly, we have (2) X is unbounded in R, R {B x : x X} = for any X R. In particular, X above is unbounded in R, R. By GFCP(κ, λ), there is X X of cardinality < λ such that X is already unbounded in R, R. Thus, again by (2), α I X α = for I = {α < δ : c α X }. (Proposition 13. )

13 Axiomatization of generic extensions... *** Fremlin-Miller Covering Property *** (13/13) Theorem 14. Let κ < µ be regular cardinals. Suppose that P {α}, α < µ are posets such that (3) P {α} = P{0} for all α < µ; (4) P = fin α<µ P α satisfies the c.c.c.; (5) P {0} κ = κ ℵ 0, κ+ < µ (e.g. κ = ℵ 1 and µ ℵ 3 under CH). Then P GFCP(µ, κ + ). Remark. The proof of the theorem above also works for corresponding sideby-side product of random forcing. Open Problems Does WFN imply HP(κ) for all regular κ ℵ 2? Does WFN (+ 2 ℵ 0 = ℵ 2 +?) imply GFCP(ℵ 3, ℵ 2 )? What is the/a reasonable axiomatization of random models? Cf.: There is an attempt to axiomatization of the iterated Sacks model by Ciesielski and Pawlikowski who devised the Covering Property Axiom CPA.