Sets and Computations - IMS

Sets and Computations
(30 Mar - 30 Apr 2015)

~ Abstracts ~

Axiomatizing some small classes of set functions
Toshiyasu Arai, Chiba University, Japan

In this talk we axiomatize the class of rudimentary functions, one of primitive recursive functions, one of safe recursive set functions, one of predicatively computable functions augmented with an iota-operator, and relativized classes of these.

Theories for feasible set functions
Arnold Beckmann, Swansea University, UK

Recently, various restrictions of the primitive recursive set functions have been proposed with the aim to capture feasible computation on sets. Amongst these are the "Safe Recursive Set Functions" (Beckmann, Buss, Friedman, accepted for publication in JSL) the "Predicatively Computable Set Functions" (Arai, Archive for Mathematical Logic, vol. 54 (2015), pp. 471-485) and the "Cobham Recursive Set Functions" (Beckmann, Buss, Friedman, Müller, Thapen, in progress).

In this talk I will discuss suitable restrictions of Kripke-Platek set theory which have the potential to capture some of these classes as their Sigma-1-definable set functions. I will develop definability in these theories, establish the above goal for the Safe Recursive Set Functions, and may end proposing a theory for the Cobham Recursive Set Functions.

Cobham recursive set functions
Sam Buss, University of California at San Diego, USA

This talk discusses a new notion of polynomial time computability for general sets, based on $\epsilon$-recursion with a Cobham style bounding using a new smash function tailored for sets. These are called the Cobham Recursive Set Functions (CRSF), and give a notion of polynomial time computability intrinsic to sets. The smash function accommodates polynomial growth rate of both the size of the transitive closure and the rank of sets. For suitable encodings of binary strings as hereditarily finite sets, the CRSF functions are precisely the usual polynomial time computable functions. We also discuss normal forms and closure properties for CRSF, and the relationship to Arai's Predicatively Computable Set Functions. This is joint work with A. Beckmann, S. Friedman, M. Mueller and N. Thapen.

Regularity properties and derived forcing properties
David Chodounský, Institute for Mathematics of the Czech Academy of Sciences, Czech Republic

I will define regularity properties of subsets of complete Boolean algebras, and introduce derived chain condition-like and properness-like forcing properties. These forcing properties have nice behavior and allow fairly general treatment. The prime example of a regularity property is 'being a centered set,' and the corresponding forcing properties are called Y-c.c. and Y-properness. The talk will focus on the general approach. This is joint work with Jindra Zapletal.

Generic I0 at $\aleph_\omega$
Vincenzo Dimonte, Kurt Gödel Research Center for Mathematical Logic, Austria

It is common practice to consider the generic version of large cardinals defined with an elementary embedding, but what happens when such cardinals are really large? The talk will concern a form of generic I0 and the consequences of this extravagant hypothesis on the "largeness" of the powerset of $\aleph_\omega$. This research is a result of discussions with Hugh Woodin.

Ideals, almost disjoint refinements, and mixing reals
Barnabas Farkas, Kurt Gödel Research Center for Mathematical Logic, Austria

With Y. Khomskii and Z. Vidnyanszky, we investigated families of subsets of omega which have almost disjoint refinements (ADR's) with respect to given ideals. I am going to talk about the following topics: 1) The existence of perfect almost disjoint families of positive sets with respect to given ideals. 2) New examples of projective ideals. 3) The existence of ADR's of families of positive sets with respect to everywhere meager (e.g. analytic and coanalytic) ideals. 4) The existence of ADR's of the family of all positive sets from the ground model in forcing extensions. 5) A new type of peculiar reals forcing notions can add, and their connections to classical properties of forcing notions.

The spectrum of $\kappa$-maximal cofinitary groups
Vera Fischer, Kurt Gödel Research Center for Mathematical Logic, Austria

For $\kappa$ an arbitrary regular uncountable cardinal we define the spectrum of $\kappa$-maximal cofinitary groups (abbreviated $\kappa$-mcg) as the set of all possible cardinalities of $\kappa$-mcg's.

Generalizing Blass's result on the spectrum of mad families on $\omega$, we provide sufficient conditions for a closed set of cardinals to be generically realized as the spectrum of $\kappa$-mcgs. We will conclude with some current directions of research and open questions.

Some cardinals on the right side of Cichon's diagram
Arthur James Fischer, University of Vienna, Austria

In a joint paper with Martin Goldstern, Jakob Kellner and Saharon Shelah we construct models of set theory in which the following string of inequalities in Cichon's diagram holds: \aleph_1 < non(M) < non(N) < cof(M) < 2^{\aleph_0}.

This is accomplished by a "creature iteration" forcing, a form of product somewhere between finite and countable support.

Reflection principles in terms of winning strategy of certain infinite games
Sakae Fuchino, Kobe University, Japan

We present several set theoretic principles claiming the existence of winning strategy (for a player) in certain infinite games which are variants of the games introduced by Philipp Doebler. These principles are proved to be consequences of infinite-combinatorial reflection principles like Rado's conjecture or Fleissner's Axiom R and they imply Fodor-type Reflection Principle (which is not the case by the game theoretic reflection principle introduced by Doebler). The relationship between these principles and some other set theoretic assertions are studied.

Many cardinals on the left side of Cichon's diagram
Martin Goldstern, Technische Universität Wien, Austria

In a joint paper with Diego Mejia Guzman and Saharon Shelah, we construct a model of set theory where the following cardinals are all distinct and also distinct from aleph_1 and continuum:
-additivity of measure, covering of measure,
-the bounding number b, and the uniformity of the meager ideal.
We use a finite support iteration of "small" forcing notions. The main problem is to show that the forcing notion that adds an eventually different real (and thus increases the uniformity number) will not destroy an unbounded family.

Limit-computable categoricity of computable structures
Valentina Harizanov, George Washington University, USA

We investigate computability-theoretic categoricity of a computable structure A. Such A is computably categorical if for every computable isomorphic B there is a computable isomorphism. The structure A is relatively computably categorical if for every isomorphic B there is an isomorphism computable relative to the atomic diagram of B, or, equivalently, A has a computably enumerable Scott family of existential formulas. Computable categoricity and relative computable categoricity often, but not always coincide for structures from natural classes. We similarly define categoricity and relative categoricity corresponding to higher levels in hyperarithmetic hierarchy. Here we focus on limit-computable categoricity versus relative limit-computable categoricity. Recent work is joint with E. Fokina, S. Friedman, and D. Turetsky.

Computable structures relative to a cone
Matthew Harrison-Trainor, University of California at Berkeley, USA

Computable structures can often be constructed to exhibit pathological behaviour. Structures which naturally occur in mathematics are often more nicely behaved and do not have such pathological behaviour. One can study the properties of natural structures by relativizing to a cone of Turing degrees; if a natural structure satisfies some property $P$ on a cone, we would expect it to satisfy the unrelativized property $P$. We will consider two examples of such properties. We will begin with degree spectra of relations and we will give some results towards a classification of the possible degree spectra relative to a cone. We will also consider degrees of categoricity relative to a cone. We will show that, relative to a cone, every structure has a strong degree of categoricity and that this degree is an iterate of the jump.

"Degree spectra on a cone" for polish spaces
Takayuki Kihara, Japan Advanced Institute of Science and Technology, Japan

The notion of degree spectrum of a structure in computable model theory is defined as the collection of all Turing degrees of presentations of the structure. We introduce the degree spectrum of a represented space as the class of collections of all Turing degrees of presentations of points in the space. We see that the degree spectrum on a cone connects descriptive set theory, topological dimension theory and computability theory.

Through this new connection, we construct continuum many infinite dimensional Cantor manifolds possessing Haver's property C whose Borel structures at an arbitrary finite rank are mutually non-isomorphic. This solves Motto Ros' problem on linear isometric classification of Banach spaces of bounded real valued Baire n functions on Polish spaces, and also strengthen various theorems in infinite dimensional topology such as Roman Pol's solution to Pavel Alexandrov's old problem.

To prove our main theorem, an invariant which we call degree co-spectrum, a collection of Turing ideals realized as lower Turing cones of points of a Polish space, plays a key role. The key idea is measuring the quantity of all possible Scott ideals (omega-models of WKL) realized within the degree co-spectrum (on a cone) of a given space.This is joint work with Arno Pauly.

Fragments of Kripke-Platek set theory and the metamathematics of $\alpha$-recursion theory
Wei Li, Kurt Gödel Research Center for Mathematical Logic, Austria

Fragments of the Kripke-Platek set theory deal with structures of set theory without full foundation. This is a joint project with Sy Friedman and Tin Lok Wong from the Kurt Gödel Research Center. One motivation of this project is the connection between nonstandard arithmetic and $\alpha$-recursion theory. $\alpha$-recursion theory has its generalized arguments in $L_\alpha$'s, where $\alpha$ is a standard ordinal. And in nonstandard arithmetic, we work in models without full induction.

Induction is the dual of foundation. In their developments, techniques and results have many overlaps. Yet, reasons for this phenomenon are yet to be found. In this talk, we present our study of logic strength of foundation and recursion theoretic results in nonstandard models of set theory. These models are "between" those in nonstandard arithmetic and those in $\alpha$-recursion theory. It is one attempt to search for the reason for the mysterious connections between these two areas.

Dropping polishness
Andrea Medini, Kurt Gödel Research Center, Austria

Classical descriptive set theory studies the subsets of complexity Gamma of a Polish space X, where Gamma is one of the (boldface) Borel or projective pointclasses. However, the definition of a Gamma subset of X extends in a natural way to spaces X that are separable metrizable, but not necessarily Polish.

When one "drops Polishness", many classical results suggest new problems in this context. We will discuss some early examples, then focus on the perfect set property. More precisely, we will determine the status of the statement. "For every separable metrizable X, if every Gamma subset of X has the perfect set property then every Gamma' subset of X has the perfect set property" as Gamma, Gamma' range over all pointclasses of complexity at most analytic or coanalytic.

Hilbert's tenth problem for subrings of the rationals
Russell Miller, Queens College, USA

For an arbitrary ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of those polynomials $p$ in $R[X_1,X_2,\ldots]$ for which $p=0$ has a solution in $R$. In 1970, Matiyasevich completed work of Davis, Putnam, and Robinson, showing that $HTP(\mathbb Z)$ is undecidable and in fact has Turing degree $\boldsymbol{0'}$. The Turing degree of $HTP(\mathhbb Q)$ remains unknown.

We will discuss the state of knowledge about $HTP(R)$ when $R$ is a subring of $\mathbb Q$. A recent result of Eisentrager, Park, Shlapentokh, and the speaker shows that every computably enumerable Turing degree between the degrees of $HTP(\mathbb Q)$ and $HTP(\mathbb Z)$ can be realized as the degree of $HTP(R)$ for such a subring $R$, and indeed that $R$ can be taken to be computably presentable. We will also consider questions about the possible asymptotic densities within $\mathbb Q$ of such subrings, and about the extent to which these methods can be extended to other subrings.

Computable structure theory and polish group actions
Antonio Montalban, The University of California at Berkeley, USA

We show how some results from Computable Structure Theory can be generalized to all computable Polish group actions.

Computable model theory over the reals: some results and problems
Andrey Morozov, SB RAS Sobolev Institute of Mathematics, Russia

The talk is a survey of results on $\Sigma$--presentable structures over HF(R) (the hereditarily finite superstructure over the ordered field of reals). Such structures could be considered as analogs of computable structures in case we have in our disposal an exact realization of the reals (R) and can compute exact values of roots of algebraic equations and then use them in further computations.

The basic problems considered in the talk will be
the existence of $\Sigma$--presentations for various structures,
the number of non $\Sigma$--isomorphic presentations,
and the existence of $\Sigma$--parameterizations for classes of presentations.
There will be also some discussion on the specific methods used in the proofs and on the differences between our case and the classical computable model theory.

The Gamma question and cardinal characteristics
Andre Nies, University of Auckland, New Zealand

A set can be seen as close to computable if asymptotically a large percentage of it equals a computable set. The Gamma value of a Turing degree is the infimum of these percentages over all sets in the degree. It is open whether this value can be properly between 0 and one half.

We study this question using a computability theory analog of cardinal characteristics. The analog of non(N) for a Turing degree b is that each Schnorr test that b computes is passed by a computable set. We start from the result that each such b is low in that its Gamma value is at least one half. We give a combinatorial characterisation of this property in terms of traces (effective slaloms). Thereafter we study analogs of the smaller characteristic cov(M). These are the "infinitely often equal" degrees.

This is joint work with Benoit Monin

Combinatorial properties and strong colorings
Yinhe Peng, Chinese Academy of Sciences, China

This is a joint work with Liuzhen. We investigated the osc map introduced by Justin Moore and found more combinatorial properties. We then found strong negative partition relations using these properties.

The topology of random graphons
Jan Reimann, Pennsylvania State University, USA

In 2010, Petrov and Vershik showed how to obtain countable universal random graphs by sampling them from continuous objects, now commonly known as graphons. Their approach was extended recently to other structures by Ackerman, Freer, and Patel.

We investigate the topological complexity of universal graphons. Our main result is that if a 0-1-valued graphon satisfies types in continuous way, then the induced fiber topology (also known as the r_W-topology) is not compact.

This is joint work with Cameron Freer.

Proof of SCH from reflection principles without using scales
Hiroshi Sakai, Kobe University, Japan

It is known that the Weak Reflection Principle and some of its variants imply the Singular Cardinal Hypothesis (SCH). Proofs of SCH from these reflection principles use the following fact on scales, due to Shelah:

Theorem (Shelah)
Assume that SCH fails, and let $\lambda$ be the least singular cardinal at which SCH fails. Then there is a better scale at $\lambda$.

This is a deep theorem in PCF theory and quite useful, but its proof is somewhat complicated. In this talk, I will present simple proofs of SCH from reflection principles, which do not use scales.

Computability theory and uncountable structures
Noah Schweber, The University of California at Berkeley, USA

Computable structure theory studies the computability-theoretic complexity of mathematical structures. For example, we can compare two countable structures A and B by asking whether every copy of B with domain $\omega$ computes a copy of B; if so, we say A is *Muchnik reducible* to B. Since Turing computability only makes sense for subsets of $\omega$, Muchnik reducibility - and computable structure theory in general - only directly applies to countable structures. However, by looking at generic extensions of the universe, we can treat structures of arbitrary cardinality. In this talk we will discuss computable structure theory in generic extensions, including connections with set theory and model theory, and present some directions for future research.

Towards the effective descriptive set theory
Victor Selivanov, A.P. Ershov Institute of Informatics Systems, Russia

We give a brief survey of effective descriptive set theory and present some (hopefully) new results. In particular, we prove effective versions of some classical results about measurable functions (in particular, any computable Polish space is an effectively open effectively continuous image of the Baire space, and any two perfect computable Polish spaces are effectively Borel isomorphic). We derive from this extensions of the Suslin-Kleene theorem, and of the effective Hausdorff theorem for the computable Polish spaces (this was recently established by Becher and Grigorieff with a different proof) and for the computable omega-continuous domains (this answers an open question from the paper by Becher and Grigorieff).

Generalized degree structures under large cardinals
Xianghui Shi, Beijing Normal University, China

The notion of Turing degree can be generalized to large ordinals, in particular to uncountable cardinals. Sy Friedman showed that in Jensen's constructible universe, the generalized degree structures at singular cardinals of uncountable cofinality is eventually well ordered. Recent developments show that interesting new degree structures appear at singular cardinals of countable cofinality in core models for some large cardinals. In this talk, I will present some generalized degree structures and discuss the correlation between the complexity of degree structures and the strength of relevant large cardinals.

On the jumps of the degrees below an r.e. degree
Richard Shore, Cornell University, USA

A perhaps plausible conjecture about distinguishing among some r.e. degrees based on the jumps of degrees below them would prove the rigidity of the r.e. degrees. We show that the conjecture is false by exhibiting, for every $\mathbf{c}$ r.e. in and above $\mathbf{0}^{\prime}$, distinct degrees $\mathbf{a}$ and $\mathbf{b}$, each with jump $\mathbf{c}$, such that the classes of the jumps of degrees below $\mathbf{a}$ and below $\mathbf{b}$ are the same.

Jjoint work with David Belanger.

Prikry forcing and square properties
Dima Sinapova, University of Illinois at Chicago, USA

Prikry type forcing is the standard way of constructing models of the failure of SCH. On the other hand not SCH is at odds with the failure of the weaker square principles. I will go over some consistency results about not SCH and failure of squares. Then I will present a dichotomy theorem characterizing what type of Prikry posets add weak square. This is joint work with Spencer Unger.

On normal numbers
Theodore Slaman, The University of California at Berkeley, USA

A real number is simply normal in base b if in its base-b expansion each digit appears with asymptotic frequency 1/b. It is normal in base b if it is simply normal in all powers of b, and absolutely normal if it is simply normal in every integer base. By a theorem of E. Borel, almost every real number is absolutely normal.

We give an efficient algorithm, which runs in nearly quadratic time, to compute the binary expansion of an absolutely normal number. We demonstrate the full logical independence between normality in one base and another. We give a necessary and sufficient condition on a set of natural numbers M for there to exist a real number X such that X is simply normal to base b if and only if b is an element of M. We show the independence between a real number's exponent of irrationality and the set of bases to which it is simply normal.

This is joint work with Verónica Becher and, in part, with Pablo Heiber and Yann Bugeaud.

Partial homogeneity of projective Fraisse limits and homogeneity of the pseudo-arc
Slawomir Solecki, University of Illinois at Urbana-Champaign, USA

The pseudo-arc is the generic compact connected subset of the plane (or any Euclidean space of dimension bigger than 1 or the Hilbert cube). By a fundamental result of Bing, it is homogeneous as a topological space. By work of Irwin and myself, the pseudo-arc is represented as a quotient of a projective Fraisse limit, which makes it possible to discretize a continuous situation.

In this joint work with Tsankov, we determine partial homogeneity of the projective Fraisse limit, which involves combinatorial and "projective" model theoretic arguments (e.g., a notion of dual type). Further, we prove a transfer theorem, through which we recover Bing's result from our partial homogeneity. Time permitting, I will comment on the generality of the method and on connections with descriptive set theoretic complexity of Borel equivalence relations.

Universal functions and universal graphs
Juris Steprans, York University, Canada

A function of two variables F(x,y) was defined by Sierpinski to be universal if for every other function G(x,y) there exist functions h(x) and k(y) such that G(x,y)=F(h(x),k(y)). Various aspects of this question were examined in a paper LMSW (Larson, Miller, Steprans and Weiss). While the universality of Sierpinski seems similar to model theoretic universality, there is a key difference in the role played by the range of the function in the two cases. This was the motivation for the following question asked in LMSW: Does the existence of a universal graph of cardinality aleph_1 imply the existence of a universal colouring of the complete graph on omega_1 with countably many colours? I will discuss joint work with S. Shelah providing a negative answer to this question.

The bounding number and the Ramsey calculus below the continuum
Stevo Todorcevic, Université Paris Diderot, University of Toronto, Canada

We shall examine the Ramsey theoretic statement $$\gamma\rightarrow (\gamma, \alpha )^2$$ for cardinals $\gamma$ not bigger than the continuum. The special case of the bounding number $\matfrak{b}$ will be discussed.

Definable maximal orthogonal families in the Sacks extension of L
Asger Toernquist, University of Copenhagen, Denmark

Two Borel probability measures nu and mu on Cantor space are orthogonal if there is a Borel set which has measure 1 for nu, but measure 0 for mu. An orthogonal family of measures is a family of pairwise orthogonal measures; it is maximal if it is maximal under inclusion.

Some years ago, Vera Fischer and I showed that in L there is a Pi-1-1 (lightface) maximal orthogonal family (a "mof") of measures in L, but that adding a Cohen real to L destroys all Pi-1-1 mofs; subsequently, it was shown that the same holds if we add a random real (Friedman-Fischer-T.).

This motivated the question: Can a Pi-1-1 mof coexist with a non-constructible real? In this talk, we answer this by showing that there is a Pi-1-1 mof in the Sacks extension of L.

(1) there exists a regular Lindel\"of P-space of pseudocharacter $\le \omega_1$ and of size $>2^\omega$,

(2) CH and $\square(\omega_2)$ hold, or (3) CH holds and there exists a Kurepa tree.

Then there exists a regular Lindel\"of space with points $G_\delta$ and of size $>2^\omega$. This shows that the non-existence of such a Lindel\"of space with CH is a large cardinal property.

Definable semifilters and N*
Jonathan Verner, Charles University, Czech Republic

I will show how relatively simple generalizations of theorems from the theory of ultrafilters on N to ultrafilters on semifilters yield interesting applications in dynamical systems and combinatorics. In particular I will show that there are idempotents in N* which are both minimal and maximal thus answering an open question of Hindmann and Strauss. The results are joint work with Will Brian.

Ordinal notation systems and well partial orders
Andreas Weiermann, Ghent University, Belgium

We discuss ordinal notation systems and their relation to well partial orderings. In quite a few times ordinal notation systems appear as maximal order types of underlying well partial orders. This led to a general conjecture about the maximal order type of tree-based well partial orderings. This conjecture could be verified in many cases but our PhD student J. Van der Meeren found one situation in which the conjecture turned out te be false. In our presentation we give a survey about these phenomena which have been obtained in joint work with M. Rathjen and J. Van der Meeren.

Continuous transfinite Blum-Shub-Smale computations and a church-like thesis for poly-time on omega-strings
Philip Welch, University of Bristol, UK

It is natural to think of poly-time computation on omega-length strings as those to be completed by some stage omega^n, if one is considering transfinite discrete computational models. The obvious example is that of restricting Infinite time Turing machines to less than omega^omega steps of computation. Several of such models yield the same class of functions, and we prove the equivalence of such for BSS machines. This class then coincides with the Safe Recursive Set Functions of Beckman,Buss and Friedman. We argue that no 'effective' method on such strings in such times, could deliver anything different.

Pure patterns and ordinal numbers
Gunnar Wilken, Okinawa Institute of Science and Technology, Japan

I will present some of my work on pure patterns of resemblance and their relationship with classical ordinal notations.

The arithmetized completeness theorem
Tin Lok Wong, Kurt Gödel Research Center for Mathematical Logic, Austria

The Arithmetized Completeness Theorem is a formalization of Goedel's Completeness Theorem within arithmetic. It has a wide range of applications in the model theory of first- and second-order arithmetic. In the talk, I will present a number of such classical applications, and discuss generalizations of these by Ali Enayat (Gothenburg) and me to models with only bounded collection and exponentiation.

Tower of master conditions
Liuzhen Wu, Chinese Academy of Sciences, China

We will illustrate an approach to obtain generic condition for towers of models using master conditions. As an example, we describe how to force the joint precipitousness of nonstationay ideal and tail club guessing ideal on omega_1 from optimal large cardinal hypothesis.

Strengthened Ramsey's theorem, finitary Ramsey's theorem and their iteration
Keita Yokoyama, Tokyo Tech University, Japan

In this talk, we will consider two versions of Ramsey's theorem in the setting of reverse mathematics. One is a generalization infinite Ramsey's theorem combining with weak Koenig's lemma, and the other is a generalization of Paris-Harrington principle. Although they come from different idea, they are actually equivalent. Then, we consider their iterated versions and study their strength.

Todorcevic's fragments of Martin's axiom and a recent result due to Bagaria and Shelah
Teruyuki Yorioka, Shizuoka University, Japan

In the recent article [On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom, arXiv:1502.05500 ], Joan Bagaria and Saharon Shelah proved that it is consistent that MA(sigma-centered) holds, every Aronszajn tree is special, and MA(sigma-linked) fails. This answers the question by Bagaria in his paper [Fragments of Martin's axiom and $\Delta^1_3$ sets of reals. Ann. Pure Appl. Logic 69 (1994), no. 1, 1-25].

Stevo Todorcevic introduced Ramsey theoretic statements which are implied by Martin's Axiom in 1980s. One of them is the following: For an integer $n \geq 2$, K'_n is the statement that every ccc partition on $[ \omega_1 ]^n$ has an uncountable homogeneous set. It is known that Martin's Axiom implies every K'_n and K'_{n+1} implies K'_n. There are a couple of questions about implications between them.
One of them is the following:
Does K'_n imply K'_{n+1}?
In particular, does K'_3 imply K'_2?
I will talk some results about these questions using the technique due to Bagaria and Shelah.

On the reals weakly low for $K$
Liang Yu, Nanjing University, China

We shall take a further look at the reals weakly low for $K$. This is a joint work with Wolfgang Merkle.

Group actions and countable dense homogeneous spaces
Shuguo Zhang, Sichuan University, China

In joint work with JIAKUI YU and LYUBOMYR ZDOMSKYY, we show that if an analytic group G makes X wCDH, then X is a CDH polish space. As a corollary, we show that in a locally compact space, the CDH, the wCDH, the SCDH and the omegaA are equivalent, which answer questions of van Mill. We also show that for a filter F on omega, F has separation property if and only if it is a non-meager subset of the Cantor space.

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