TOPICS IN SET THEORY: Example Sheet 2

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1 TOPICS IN SET THEORY: Example Sheet 2 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge Michaelmas Dr Oren Kolman ok261@dpmms.cam.ac.uk 1 (i) Suppose that x and y belong to the class W F of well-founded sets. Find bounds for the ranks of the following sets in terms of the ranks of x and y : x, P (x), {x}, x y, x y, x y, {x, y}, x, y, and y x. (ii) Calculate the ranks of the sets N, Z, Q, R, and C. 2 Prove the basic properties of the hierarchy {V α : α Ord}. (i) α β V α V β. (ii) α < β V α V β. (iii) V α is a transitive set: x V α x V α. (iv) V α = {x W F : rank(x) < α}, where rank(x) = min{β : x V β+1 }. (v) y x rank(y) < rank(x). (vi) rank(α) = α. (vii) Ord V α = α. (viii) For every n ω, V n < 0 V ω = 0 (AC) for all α Ord, V ω+α = ℶ α. 3 (i) Suppose that M is a transitive class. Show that ZF C Extensionality M and ZF C Regularity M. (ii) For a L( ) -formula ϕ(x, y) with free variables x, y, y 1,..., y n, and a variable z not free in ϕ(x, y), let ( ) ϕ(x,y) be the assertion ( x a)(!y)ϕ(x, y) z y(y z x(x a ϕ(x, y))). The notation!wψ abbreviates the formula z w(ψ w = z), where z is the first variable different from w and not free in ψ. Show that the schema ( ) ϕ(x,y) is an equivalent form of the schema of the Axiom of Replacement. (iii) Prove that the Reflection Principle for {V α : α Ord} implies the Axiom schema of Replacement. [Hint: Suppose a V α referring to part (i), reflect to a V β, and use the Axiom of Separation to find the right candidate for the image of a.] Comment: sometimes it may prove less onerous to check the Reflection Principle instead of Replacement, e.g., when showing that the class L is an inner model of ZF C, it is equivalent to check that L is an inner model of Levy-Montague set theory LM. 1

2 4 (i) Assume the Axiom of Choice. Suppose I is a non-empty set and for each i I, λ i is an infinite cardinal. Show i I λ i I sup i I λ i, where I is the cardinality of I. [Hint: enumerate λ i and think up a surjection from I λ onto i I λ i where λ = sup i I λ i.] (ii) Suppose κ is an infinite cardinal. Prove that the (cardinal) successor κ + κ is a regular cardinal. [Hint: part (i).] of 5 (i) Prove that for every infinite cardinal κ, 2 κ = κ κ. (ii) Let κ, λ i (i I) be infinite cardinals. Prove: (a) κ i I λ i = i I κλ i (b) ( i I λ i) κ = i I λκ i. 6 Suppose α and β are ordinals. Prove at least (iv) -(vii) of the following: (i) α+1 < 2 2α α β implies β α = 2 β. (ii) 0<n<ω n = 2 0 n<ω n = 0 ω α<ω+ω α = 0 ω+ω. (iii) α < β implies β α+1 = β α α+1. (iv) GCH implies α β = α if β < cf( α ) α+1 if cf( α ) β α β+1 if α β. (v) ω1 ω = 0 ω 2 ω 1. (vi) α < ω 2 implies 2 α = 1 α 2 2 (vii) 2 0 > ω implies 0 ω = 2 0 (viii) 2 1 = 2 and 0 ω > ω1 implies 1 ω1 = 0 ω. (ix) 2 0 ω1 implies )ג ω ) = 2 0 and )ג ω1 ) = 2 1. (x) = 2 implies ω ω1. 7 (i) Prove that an infinite cardinal κ is a strong limit cardinal if and only if κ = ℶ δ for some limit ordinal δ. (ii) Let κ be a limit cardinal and λ > 0. Let δ be a limit ordinal such that λ < cf(δ). Suppose that {κ ξ < κ : ξ < δ} is a strictly increasing sequence of cardinals such that κ = ξ<δ κ ξ. Show that κ λ = ξ<δ κ ξ λ. 2

3 (iii) Prove that if δ > 0 is a limit ordinal, then cf(ℶ δ ) = cf(δ). 8 (i) Suppose κ is a limit cardinal and λ < cf(κ). Prove κ λ = α<κ α λ. (ii) Suppose κ is a limit cardinal and λ cf(κ). Prove κ λ = (sup α<κ α λ ) cf(κ). (iii) Suppose κ > cf(κ) is not a strong limit cardinal. Prove κ <κ = 2 <κ > κ. (iv) Suppose κ > cf(κ) is a strong limit cardinal. Prove 2 <κ = κ and κ <κ = κ cf(κ). 9 (i) Show that GCH implies SCH. (ii) Show that the Gimel function ( κ )ג determines the class function (κ, λ) κ λ. (iii) Find a consistent assignment of cardinals to 2 n for n < ω such that )ג ω ) < 2 ω. What does this imply about the relation between SCH and GCH? 10 Absoluteness results All formulas and terms are in the vocabulary of ZF C unless otherwise indicated. Suppose that ϕ, ϕ 1, and ϕ 2 are absolute and T is a fragment of ZF C. (i) Suppose that ψ is a formula with the same free variables as ϕ such that T ϕ ψ. Show that ψ is absolute for transitive models of T. (ii) Suppose that yϕ 1, zϕ 2 and ψ have the same free variables suppose T yϕ 1 ψ and T zϕ 2 ψ. Show that ψ is absolute for transitive models of T. (iii) Suppose that ψ(y) and the term t are absolute for transitive models of T. Show that ψ(t) is absolute for transitive models of T. (iv) Suppose that t is absolute. Show that x t and t x are absolute. (v) Suppose that ψ(y) and the term t are absolute for transitive models of T. Show that {y t : ψ(y)} is absolute for transitive models of T. (vi) Prove that the following terms and predicates are absolute for transitive models of ZF (in each instance, it suffices to produce a ZF -provably equivalent absolute or 0 -formula): y x z = {x, y} z = {x} z = x, y z = x z = x y z = x y z = x \ y z = x y f is a function y = dom(f) y = range(f) y = y = s(x), where s(x) = x {x} y = 1 y = 2 y = f x (where f x means the application of f to x ) y = f x x is transitive x Ord (remember that Foundation is an axiom of ZF C ) x is a limit ordinal x = ω. 3

4 (vii) Suppose that the term s(y, z) is absolute, and ZF C αt(α) = s(t α, α). Show that the formula y = t(α) is absolute. [For a formula ϕ(x), ϕ(α) abbreviates α Ord ϕ(α).] (viii) Prove that the following ordinal-defined operations are absolute: α + β α β α β rank(x) y is the transitive closure of x. (ix) Determine which of the following are (a) absolute, (b) absolute for V κ, when κ is a strongly inaccessible cardinal (brief explanations suffice): Z, (Q, ), (R, ), x is countable, y = P (z), α is a cardinal, 1, T is a Suslin tree, y = δ, δ = δ, z = ℶ ξ, x L α, ( α)(x L α ), ZF C ϕ Optional: ( κ )ג, M is an R -module over the commutative ring R, P = NP, the real Hilbert space l 2, X is a complex Banach space, Y is an inseparable topological space, the Singular Cardinals Hypothesis, the Riemann Hypothesis. 11 (i) Let ZF C be the first-order theory whose axioms are obtained from ZF C by omitting the axiom of infinity. Show that (*) (V ω, (V ω V ω )) is a model of ZF C. (ii) Show that every element of V ω is finite. Deduce that the axiom of infinity cannot be proved from the other axioms of ZF C. (iii) Can the assertion (*) be proved from ZF C? Explain. 12 (i) Suppose that M is a transitive model of ZF (or a large enough finite fragment of ZF including the Power Set Axiom) and let x M. Prove that P (x) M = P (x) M. Deduce that the Power Set Axiom holds in M if and only if x M y M(P (x) M y). (ii) Suppose that V α reflects (a large enough finite fragment T of) ZF C, let β < α. Prove that V Vα β = V β. Hence complete the second proof that neither ZF nor ZF C is finitely axiomatizable. 13 (i) Suppose there exists an ordinal α such that V α is a model of ZF C. Show that the least such ordinal α has cofinality ω. (ii) Suppose κ is a strongly inaccessible cardinal. Prove that V κ ZF C. (iii) Deduce that the converse of part (ii) is false. is a model of 14 The Reckless Axiom at κ asserts: (RA κ ) for every partial order P and family D of κ dense open sets in P, there exists a D -generic filter G. Refute RA 1. Open Research Problems. 4

5 (i) Suppose that n has the tree property for 1 < n < ω. Does ω+1 have the tree property? (ii) Can every regular cardinal greater than 1 have the tree property? See: James Cummings, Notes on singular cardinal combinatorics, Notre Dame J. Formal Logic 46 (2005), no. 3,