Violating the Singular Cardinals Hypothesis Without Large Cardinals

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1 Violating the Singular Cardinals Hypothesis Without Large Cardinals Talk at the University of Bristol by Peter Koepke (Bonn), joint work with Moti Gitik (Jerusalem) November 29, 2010 Cantor s Continuum Hypothesis Georg Cantor proved: Theorem 1. The power set {x x N} of N is not denumerable. In the language of cardinal arithmetic this reads: Theorem 2. 2 ℵ 0 ℵ 1. Cantor conjectured Conjecture 3. (Cantor s Continuum Hypothesis, CH) 2 ℵ 0 =ℵ 1. Kurt Gödel proved the consistency of CH, assuming the consistency of the Zermelo- Fraenkel axioms ZFC, by constructing the model L of constructible sets Theorem 4. L CH. Paul Cohen proved the opposite relative consistency Theorem 5. Any (countable) model M of ZFC can be extended to a model M[G] of ZFC+2 ℵ 0 >ℵ 1. For this, Cohen invented the method of forcing to adjoin further objects G to M. 1

2 G is a (generic) limit of approximations (conditions) in some partial order. For the CH construction we want to adjoin a characteristic function F satisfying 1. F:λ ω 2 for some λ ℵ 2 M 2. i<j<λ(λn.f(i,n) λn.f(j,n) A i A j ω A i A j F 0 1 λ i j Cohen s partial order for this is essentially P ={p n<ω D [λ] <ω p:d n 2} partially ordered by reverse inclusion: p q (p is stronger than q) iff p q 2

3 This may be pictured as ω 0 1 p q q i j λ If G P is a generic path through P then F = {p p G} is as required. Hausdorff s Generalized Continuum Hypothesis Felix Hausdorff conjectured an extension of CH Conjecture 6. (Hausdorff s Generalized Continuum Hypothesis, GCH) α.2 ℵ α =ℵ α+1 3

4 Since GCH holds in Gödel s model L, Theorem 7. GCH is independent of ZFC. Easton proved Theorem 8. Let E: Ord Ord be a sufficiently absolute function such that E(α)>α α<β E(α) E(β) Lim(E(α)) cof(e(α))>ℵ α Then one can construct a forcing extension M[G] such that α(ℵ α is regular 2 ℵ α =ℵE(α) ) The Singular Cardinal Hypothesis is the statement (SCH) if κ is a singular strong limit cardinal then 2 κ =κ + Moti Gitik and Bill Mitchell showed Theorem 9. The following two theories are equiconsistent: ZFC+ SCH ZFC + there are many measurable cardinals SCH Without the Axiom of Choice Theorem 10. The following theories are equiconsistent: ZF 4

5 ZF + GCH holds below ℵ ω some fixed big ordinal α + there is a surjection from P(ℵ ω ) onto ℵ α, for This is a strong surjective failure of SCH, without requiring large cardinals. Injective failures possess much higher consistency strengths. The forcing Fix a ground model V of ZFC+GCH and let λ=ℵ α be some regular cardinal in V. The forcing P 0 =(P 0,, ) adjoins one Cohen subset of ℵ n+1 for every n<ω. P 0 ={p (δ n ) n<ω ( n<ω:δ n [ℵ n,ℵ n+1 ) p: n<ω [ℵ n,δ n ) 2)}. The forcing (P, P, ) is defined by P ={(p,(a i,p i ) i<λ ) (δ n ) n<ω D [λ] <ω ( n<ω:δ n [ℵ n,ℵ n+1 ), p : n<ω [ℵ n,δ n ) 2 2, i D:p i : n<ω [ℵ n,δ n ) 2 p i, i D:a i [ℵ ω \ℵ 0 ] <ω n<ω: card(a i [ℵ n,ℵ n+1 )) 1, i D(a i =p i = ))} ℵ n+1 ℵ n ℵ 2 ξ a i ℵ 1 ℵ 0 ξ p i p p j 5

6 P is partially ordered by p =(p,(a i,p i ) i<λ ) P (p,(a i,p i ) i<λ )=p iff a) p p, i<λ(a i a i p i p i ), b) i<λ n<ω ξ a i [ℵ n,ℵ n+1 ) ζ dom(p i \p i ) [ℵ n,ℵ n+1 ):p i (ζ)=p (ξ)(ζ), and c) j supp(p):(a j \a j ) i supp(p),ij a i =. equal endsegments ξ a i ξ p i p i p p p j Lemma 11. P satisfies the ℵ ω+2 -chain condition. Let G be V -generic for P. Define G = {p P (p,(a i,p i ) i<λ ) G} A = G : [ℵ n,ℵ n+1 ) 2 2 n<ω A (ξ) = {(ζ,a (ξ,ζ)) ζ [ℵ n,ℵ n+1 )}:[ℵ n,ℵ n+1 ) 2 A i = {p i (p,(a j,p j ) j<λ ) G}:[ℵ 0,ℵ ω ) 2 6

7 equal endsegments ξ a i ξ A i A A j Fuzzifying the A i Define the exclusive or function :2 2 2 by a b=0 iff a=b. For functions A,A : dom(a)=dom(a ) 2 define the pointwise exclusive or A A : dom(a) 2 by (A A )(ξ)=a(ξ) A (ξ). For functions A,A :(ℵ ω \ℵ 0 ) 2 define an equivalence relation by A A iff n<ω((a A ) ℵ n+1 V[G ] (A A ) [ℵ n+1,ℵ ω ) V). Let Ã ={A A A} be the -equivalence class of A. The symmetric submodel Set T =P(<κ) V[A ], setting κ=ℵ ω V ; 7

8 A O =(Ã i i<λ). The final model is N = HOD V [G] (V {T,A O } T i<λ Ã i ) consisting of all sets which, in V[G] are hereditarily definable from parameters in the transitive closure of V {T,A O }. Lemma 12. Every set X N is definable in V[G] in the following form: there are an - formula ϕ, x V, n<ω, and i 0,,i l 1 <λ such that X={u V[G] V[G] ϕ(u,x,t,a O V,A (ℵ n+1 ) 2,A i0,,a il 1 )}. Lemma 13. N is a model of ZF, and there is a surjection f:p(κ) λ in N. Proof. Note that for every i<λ: A i N. (1) Let i<j<λ. Then A i A j. Proof. Assume instead that A i A j. Then take n < ω such that v = (A i A j ) [ℵ n+1, ℵ ω ) V. The set D={(p,(a k,p k ) k<λ ) ξ [ℵ n+1,ℵ ω )(ξ dom(p i ) dom(p j ) v(ξ) p i (ξ) p j (ξ))} V is readily seen to be dense in P. Take (p, (a k, p k ) k<λ ) D G. Take ξ [ℵ n+1, ℵ ω ) such that ξ dom(p i ) dom(p j ) v(ξ) p i (ξ) p j (ξ)). Since p i A i and p j A j we have v(ξ) A i (ξ) A j (ξ) and v (A i A j ) [ℵ n+1,ℵ ω ). Contradiction. qed(1) Thus f(z)= { i, if z Ã i ; 0, else; is a well-defined surjection f: P(κ) λ, and f is definable in N from the parameters κ and A O. 8

9 Approximating N Lemma 14. Let X N and X Ord. Then there are n<ω and i 0,,i l 1 <λ such that V X V[A (ℵ n+1 ) 2,A i0,,a il 1 ]. Proof. Let X ={u Ord V[G] ϕ(u,x,t,a O V,A (ℵ n+1 ) 2,A i0,,a il 1 )}. By taking n sufficiently large, we may assume that For j<l set j<k<l m [n,ω) δ [ℵ m,ℵ m+1 ): A ij [δ,ℵ m+1 ) A ik [δ,ℵ m+1 ). a ij ={ξ m n δ [ℵ m,ℵ m+1 ):A ij [δ,ℵ m+1 )=A (ξ) [δ,ℵ m+1 )} where A (ξ)={(ζ,a (ξ,ζ)) (ξ,ζ) dom(a )}. Define X = {u Ord there is p=(p,(a i,p i ) i<λ ) P such that V p (ℵ n+1 ) 2 V A (ℵ n+1 ) 2, a i0 a i0,,a il 1 a il 1, p i0 A i0,,p il 1 A il 1, and p ϕ(ǔ,xˇ,σ,τ,a (ℵˇn+1) 2,A i 0,,A i l 1 )}, where σ,τ,a,a i 0,,A i l 1 are canonical names for T,A O,A,A i0,,a il 1 resp. Then X V V[A (ℵ n+1 ) 2,A i0,,a il 1 ]. (1) X X. Proof. Straightforward. qed(1) The converse direction, X X, uses an automorphism argument. 9

10 possibly different same since A il 1 possibly different p [ℵ n,ℵ n+1 ) 2 A = p i0 p i0 p i p i p ℵ 1 2 = p 2 ℵ 1 a i0 a il 1 i supp(p) One defines an isomorphism π of P below p and below p, respectively. identities outside supp(p) corresponding grey parts are equal black extensions determined by linking ordinals = q ℵ 1 2 = q 2 ℵ 1 a i0 a il 1 i supp(p) Wrapping up Lemma 15. Let n < ω and i 0, V[A V (ℵ n+1 ) 2,A i0,,a il 1 ]., i l 1 < λ. Then cardinals are absolute between V and 10

11 Lemma 16. Cardinals are absolute between N and V, and in particular κ=ℵ ω V =ℵ ω N. Proof. If not, then there is a function f N which collapses a cardinal in V. By Lemma V 14, f is an element of some model V[A (ℵ n+1 ) 2,A i0,,a il 1 ] as above. But this contradicts Lemma 15. Lemma 17. GCH holds in N below ℵ ω. V Proof. If X ℵ n and X N then X is an element of some model V[A (ℵ n+1 ) 2,A i0,, A il 1 ] as above. Since A i0,,a il 1 do not adjoin new subsets of ℵ n we have that V X V[A (ℵ n+1 ) 2 ]. Hence P(ℵ V V n ) N V[A (ℵ n+1 ) 2 V ]. GCH holds in V[A (ℵ n+1 ) 2 ]. Hence there is a bijection P(ℵ V V V n ) N ℵ n+1 in V[A (ℵ n+1 ) 2 ] and hence in N. Discussion and Remarks To work with singular cardinals κ of uncountable cofinality, various finiteness properties in the construction have to be replaced by the property of being of cardinality < cof(κ). This yields choiceless violations of Silver s theorem. Theorem 18. Let V be any ground model of ZFC + GCH and let λ be some cardinal in V. Then there is a model N V of the theory ZF+ GCH holds below ℵ ω1 + there is a surjection from P(ℵ ω1 ) onto λ. Moreover, the axiom of dependent choices DC holds in N. 11

Violating the Singular Cardinals Hypothesis Without Large Cardinals

Violating the Singular Cardinals Hypothesis Without Large Cardinals CUNY Logic Workshop, November 18, 2011 by Peter Koepke (Bonn); joint work with Moti Gitik (Tel Aviv) Easton proved that the behavior