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) (2008 05 29 21:57 )
Axiomatization of generic extensions... *** Main References *** (2/13) [1] J. Brendle and S.F.: Coloring ordinals by reals, Fundamenta Mathematicae, 196, No.2 (2007), 151-195. [2] S.F.: A generalization of a problem of Fremlin, RIMS Kôkyûroku, No.1595 (2008), 6-13. [3] S.F., S. Geschke and L. Soukup: On the weak Freese-Nation property of P(ω), Archive for Mathematical Logic, Vol.40 (2001), 425-435. [4] S.F. and L. Soukup: More set-theory around the weak Freese-Nation property, Fundamenta Matematicae 154 (1997), 159 176.
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.
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 (M.Elekes, T. Mátrai and L.Soukup, still unpublished) If U is a covering of R by closed sets such that each x R is included in uncountably many elements of U (ℵ 1 -fold covering) then U can be partitioned into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.
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.
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.
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.
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 ).
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.
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.
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.
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. )
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.