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Bounding 2D Functions by Products of 1D Functions 3 BOUNDING 2D FUNCTIONS BY PRODUCTS OF 1D FUNCTIONS FRANC¸OIS DORAIS AND DAN HATHAWAY Abstract. Given sets X, Y and a regular cardinal µ, let Φ(X,Y,µ) be the statement that for any function f : X × Y → µ, there are functions g1 : X → µ and g2 : Y → µ such that for all (x, y) ∈ X × Y , f(x, y) ≤ max{g1(x),g2(y)}. In ZFC, the statement Φ(ω1,ω1,ω) is false. However, we show the theory ZF + “the club filter on ω1 is normal” + Φ(ω1,ω1,ω) is equiconsistent with ZFC + “there exists a measurable cardinal”. 1. Introduction The Axiom of Choice allows us to define objects of size ω1 that do not behave like objects of size ω. For infinite cardinals λ1 and λ2, we introduce a statement Φ(λ1,λ2,ω) which implies in some sense that functions from λ1 × λ2 to ω behave like functions from ω × ω to ω. In Section 2 we show in ZFC that Φ(λ,ω,ω) is true iff λ < b, where b is the bounding number. We also show that in ZF that Φ(R,ω,ω) is false. In Section 3 we show that Φ(ω1,ω1,ω) is false in ZF assuming ω1 is regular and there is an injection of ω1 into R. Thus, if ZF + “ω1 is regular” + Φ(ω1,ω1,ω) holds, then ω1 is inaccessible in every inner model that satisfies the Axiom of Choice. On the other hand consider AD, the Axiom of Determinacy. This is arXiv:1601.05454v3 [math.LO] 23 Jan 2020 an axiom which contradicts the Axiom of Choice. It imposes regularity properties on both subsets of R and on bounded subsets of Θ, which is the smallest ordinal that R cannot be surjected onto. AD implies in some sense that subsets of ω1 behave like subsets of ω. Contrasting with the Axiom of Choice, in Section 4 show that AD implies Φ(ω1,ω1,ω). In fact, we show that ZF+DC + “ω1 is a measurable cardinal” implies Φ(ω1,ω1,ω). In Section 5, we aim to show that ZF+Φ(ω1,ω1,ω) implies that there are inner models with large cardinals. We show that ZF + “the club filter on ω1 is normal” + Φ(ω1,ω1,ω) implies 1) every real has a sharp and 2) there is an inner model with a measurable cardinal. Combining 1 2 FRANC¸OISDORAISANDDANHATHAWAY this with Section 4, we have that the theory ZF + “the club filter on ω1 is normal” + Φ(ω1,ω1,ω) is equiconsistent with ZFC + “there is a measurable cardinal”. We will use the following concepts several times: Definition 1.1. An inner model is a transitive model of ZF which contains all the ordinals. L is the smallest inner model. Definition 1.2. Given a set of ordinals A ⊆ Ord, L[A] is the smallest inner model N such that A ∈ N. In general, given any inner model M and a set of ordinals A, M[A] is the smallest inner model N such that M ⊆ N and A ∈ N. If A is a set of ordinals, then L[A] satisfies the Axiom of Choice. We will need the following strengthening (see [5] page 451 for a proof): Proposition 1.3. Let M be an inner model. Let A be a set of ordinals. Then M[A] exists. If moreover M satisfies the Axiom of Choice, then so does M[A]. 1.1. Acknowledgments. We would like to thank Cody Dance and Trevor Wilson for discussions relating to this paper, especially con- cerning the behavior of sets of ordinals assuming AD. I would also like to thank Phil Tosteson who originally asked about the status of Φ(λ1,λ2,ω). 2. Definition and the Bounding Number Definition 2.1. For sets X,Y and an infinite regular cardinal µ, let Φ(X,Y,µ) be the statement that whenever f : X × Y → µ, there are functions g1 : X → µ and g2 : Y → µ such that for all (x, y) ∈ X × Y , f(x, y) ≤ max{g1(x),g2(y)}. We will mostly consider the case that X and Y are cardinals (which are ordinals). When this happens, without loss of generality λ1,λ2 ≥ µ. In this section we will consider arbitrary µ, but for the rest of the paper we will focus on the µ = ω case. Theorem 2.2. Let µ be a regular cardinal. Let X be a set. Then Φ(X,µ,µ) holds iff each family hfx : x ∈ Xi of functions from µ to µ is eventually dominated (mod <µ) by some single function. Proof. First assume every such family indexed by X can be eventually dominated by a single function from µ to µ. Fix f : X × µ → µ. Let g2 : µ → µ be such that for each x ∈ X, the set Sx := {y<µ : f(x, y) > g2(y)} has size < µ. Let g1 : X → µ be such that for each x ∈ X, g1(x) ≥ sup{f(x, y): y ∈ Sx}. The functions g1 and g2 work. BOUNDING 2D FUNCTIONS BY PRODUCTS OF 1D FUNCTIONS 3 For the other direction, assume there is a family indexed by X of functions from µ to µ that cannot be eventually dominated by a single function. Using such a family, let f : X × µ → µ be such that the functions of the form y 7→ f(x, y) cannot be eventually dominated by a single function from µ to µ. By possibly increasing the values of the function f, we may assume that whenever x ∈ X and Y ∈ [µ]µ, the set f“{x} × Y is unbounded below µ. Now, consider any g2 : µ → µ. By hypothesis, fix some x ∈ X such that g2 does not eventually dominate the function y 7→ f(x, y). That is, Sx := {y ∈ µ : f(x, y) >g2(y)} is in µ [µ] , so f“{x} × Sx is unbounded below µ. Hence, there is no α<µ such that for all y<µ, f(x, y) ≤ max{α,g2(y)}, so there can be no function g1 which works (because no value of α = g1(x) works). Every family of size µ of functions from µ to µ is eventually domi- nated by a single function. On the other hand, there exists a family indexed by µ2 of functions from µ to µ that cannot be eventually dom- inated by any single function. This gives us the following: Corollary 2.3. (ZF) Let µ be a regular cardinal. Then Φ(µ,µ,µ) holds but Φ(µ2,µ,µ) does not. Corollary 2.4. (ZF) Let µ be a regular cardinal. Assume there is no injection of µ+ into µ2. Then for any cardinal λ (that is an ordinal), Φ(λ,µ,µ) holds. Proof. Any size λ family of functions from µ to µ has at most µ distinct members, and hence the family can be eventually dominated by a single function. Recall that in ZFC, b(µ) is the cardinality of the smallest family of functions from µ to µ that cannot be eventually dominated by a single function. See [1] for more on b(ω). Corollary 2.5. (ZFC) Let µ be a regular cardinal. Let λ be a cardinal. Then Φ(λ,µ,µ) holds iff λ< b(µ). 3. First Countable Topologies By Corollary 2.3, Φ(R,ω,ω) fails in ZF, and hence so does Φ(R,ω1,ω) and Φ(R, R,ω). In this section, we will show in ZF that if ω1 is regular and there is an injection of ω1 into R, then Φ(ω1,ω1,ω) fails as well. Given a topological space X and a point x ∈ X, we call B ⊆ P(X)a neighborhood basis for x iff 1) every U ∈ B is open and contains x and 2) for every open W ∋ x, there is some U ∈ B such that U ⊆ W . We call X first-countable iff every x ∈ X has a countable neighborhood basis. 4 FRANC¸OISDORAISANDDANHATHAWAY Definition 3.1. Let X be a topological space. We call X uniformly first-countable iff there is a function F with domain X such that for each x ∈ X, F (x) is a sequence hUx,n : n<ωi such that {Ux,n : n<ω} is a neighborhood basis for x. Assuming the Axiom of Choice, a topological space is first-countable iff it is uniformly first-countable. Recall that a topological space X is T1 iff for any distinct x, y ∈ X, there is an open set which contains x but not y. Theorem 3.2. (ZF) Let X be a T1 and uniformly first-countable topo- logical space. Let Y ⊆ X and assume Φ(X,Y,ω). Then we may write Y = n<ω Yn where each Yn is a closed discrete subset of Y and Y0 ⊆ Y1 ⊆ .... S Proof. Let the function F witness that Y is uniformly first-countable, where F (x) = hUx,n : n<ωi. Assume without loss of generality that for each x ∈ X, we have Ux,0 ⊇ Ux,1 ⊇ .... Let f : X × Y → ω be defined by min{n<ω : y 6∈ U } if x =6 y, f(x, y) := x,n (0 if x = y. Thus for all distinct x ∈ X and y ∈ Y , y 6∈ Ux,f(x,y). By Φ(X,Y,ω), there are functions g1 : X → ω and g2 : Y → ω such that f(x, y) ≤ max{g1(x),g2(y)} for all distinct x ∈ X and y ∈ Y .
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