The canonical function game

The canonical function game is a game of length ω ₁ introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ² ₂ absoluteness, cardinality spectra and Π₂ maximality for H(ω ₂) relative to the Continuum Hypothesis.     

The cardinal characteristic for relative γ-sets

For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γ-set in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p(ω²) and p(ωω) in terms of definable free filters on ω which is related to the pseudo-intersection number p. We show that for every uncountable standard analytic space X that either p(X)=p(ω²) or p(X)=p(ωω). We show that the following statements are each relatively consistent with ZFC: (a) p=p(ωω)<p(ω²) and (b) p<p(ωω)=p(ω²)     

Projective mad families

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω (resp. functions from ω to ω) together with b=2^ω=ω₂.     

Countable partition ordinals

The structure of ordinals of the form ω^ωβ for countable β is studied. The main result is:

Theorem 1. Ifβ<ω₁is the sum of one or two indecomposable ordinals, then

ω^ωβ→(ω^ωβ,3)².     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω₁² that is an HFD with the following property

(P)

the projection of H onto every partial product I² for I∈ω[ω₁] is onto.

Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ², there is an HFD topological group in ω₁² which has property (P).     

Global singularization and the failure of SCH

We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ≥κ⁺ if there is an embedding j:V→M with critical point κ such that H(μ)^V is included in M and j(κ)>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V^∗ where F is realised on all V-regular cardinals and moreover, all F(κ)-hypermeasurable cardinals κ, where F(κ)>κ⁺, with a witnessing embedding j such that either j(F)(κ)=κ⁺ or j(F)(κ)≥F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2^α∈{α⁺,α⁺⁺} for every cardinal α below κ (in this case every κ⁺⁺-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F)(κ)≥F(κ) or j(F)(κ)=κ⁺).     

Cascades,order, and ultrafilters

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. This allows us to prove (ZFC) that the class of strict _Jωω$J_{{\omega }^{\omega }}$-ultrafilters, introduced by J.E. Baumgartner in [2], is empty. We translate the result to the language of _<∞${<}_{\infty }$-sequences under an ultrafilter, investigated by C. Laflamme in [17], and we show that if there is an arbitrary long finite _<∞${<}_{\infty }$-sequence under u, then u   is at least a strict _Jωω+1$J_{{\omega }^{\omega +1}}$-ultrafilter.     

Wide scattered spaces and morasses

We show that it is relatively consistent with ZFC that ω² is arbitrarily large and every sequence s=〈sα:α<ω₂〉 of infinite cardinals with sα?ω² is the cardinal sequence of some locally compact scattered space.     

Mathias–Prikry and Laver–Prikry type forcing

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martin?s number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.     

Selective ultrafilters and homogeneity

We develop a game-theoretic approach to partition theorems, like those of Mathias, Taylor, and Louveau, involving ultrafilters. Using this approach, we extend these theorems to contexts involving several ultrafilters. We also develop an analog of Mathias forcing for such contexts and use it to show that the proposition (considered by Laver and Prikry) “every non-trivial c.c.c. forcing adjoins Cohen-generic reals or random reals” implies the non-existence of P-points. We show that, in the model obtained by Lévy collapsing to ω all cardinals below a Mahlo cardinal ;, any countably many selective ultrafilters are mutually generic over the Solovay (Lebesgue measure) submodel. Finally, we show that a certain natural group of self-homeomorphisms of βω-ω, chosen so as to preserve selectivity of ultrafilters, in fact preserves isomorphism types.     

Pseudocompactness of hyperspaces

We consider the following question of Ginsburg: Is there any relationship between the pseudocompactness ofXωand that of the hyperspaceX²? We do that first in the context of Mrówka-Isbell spaces Ψ(A) associated with a maximal almost disjoint (MAD) family A on ω answering a question of J. Cao and T. Nogura. The space Ψω(A) is pseudocompact for every MAD family A. We show that

(1)

(p=c) ²Ψ(A) is pseudocompact for every MAD family A.

(2)

(h<c) There is a MAD family A such that ²Ψ(A) is not pseudocompact.

We also construct a ZFC example of a space X such that Xω is pseudocompact, yet X² is not.     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

A combinatorial result related to the consistency of New Foundations

We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types (TST), TST₄. The result says that if A=〈A₀,A₁,A₂,A₃〉 is a standard transitive and rich model of TST₄, then A satisfies the 〈0,0,n〉-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations (NF). The result is a weak form of the combinatorial condition (existence of ω-extendible coherent triples) that was shown in Tzouvaras (2007) [5] to be equivalent to the consistency of NF. Such weak versions were introduced in Tzouvaras (2009) [6] in order to relax the intractability of the original condition. The result strengthens one of the main theorems of Tzouvaras (2007) [5, Theorem 3.6] which is just equivalent to the 〈0,0,2〉-property.     

The Tukey Order and Subsets of <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript>

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ? : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by \(\mathcal {K}(X)\) the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of \(\mathcal {K}(S)\) corresponding to various subspaces S of ω ₁, their Tukey invariants, and hence the Tukey relations between them. It is shown that ω ^ω is a strict Tukey quotient of \({\Sigma }(\omega ^{\omega _{1}})\) and thus we distinguish between two Tukey classes out of Isbell’s ten partially ordered sets from (Isbell, J. R.: J. London Math Society 4(2), 394–416, 1972). The relationships between Tukey equivalence classes of \(\mathcal {K}(S)\), where S is a subspace of ω ₁, and \(\mathcal {K}(M)\), where M is a separable metrizable space, are revealed. Applications are given to function spaces.     

On a σ-ideal of compact sets

We recall from [T. Mátrai, Kenilworth, Proc. Amer. Math. Soc. 137 (3) (2009) 1115-1125] a Gδσ-ideal of compact subsets of ω² and prove that it is not Tukey reducible to the ideal . This result answers a question of S. Solecki and S. Todor?evi? in the negative.     

Linear orderings and powers of characterizable cardinals

We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ⁺. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵ_α, for some α<ω₁ (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ⁺ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2^λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κ^λ>κ, then κ^λ is also characterizable (Theorem 5.4) and (2) if ℵ_α and κ^ℵ_α are characterizable cardinals, then the same is true for κ^ℵ_α+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2^ℵ_α is a characterizable cardinal, ℵ_α is characterizable by a densely ordered structure and ℵ_α is the least cardinal such that κ^ℵ_α>κ, then for all β<α+ω₁, κ^ℵ_β is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κ^ℵ_α is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].