# Blog Archives

# Topic Archive: set theory

# Normal Measures and Strongly Compact Cardinals

I will discuss the question of the possible number of normal

measures a non-kappa + 2 strong strongly compact cardinal kappa

can carry. This is part of a joint project with James Cummings.

# Professor Arthur W. Apter promoted to Distinguished Professor

Professor Arthur W. Apter, a long-standing and prominent member of the New York logic community, has been promoted to Distinguished Professor at CUNY, effective February 1, 2014.

Professor Apter is known internationally for his foundational early work in choiceless set theory and also for his work in the area of forcing and large cardinals, including especially a large body of results concerning the indestructibility phenomenon of large cardinals and the level-by-level agreement between strong compactness and supercompactness, among many other topics. A prolific researcher, he has published well over 100 articles in refereed research journals.

From his profile at the CUNY Distinguished Professor page:

Professor Arthur W. Apter was born and raised in Brooklyn, New York, where he attended New York City public schools. After graduation in 1971 from Sheepshead Bay High School, he attended MIT, where he earned his B.S. in Mathematics in 1975 and his Ph.D. in Mathematics in 1978. After spending one additional postdoctoral year at MIT, he spent two years in the Mathematics Department of the University of Miami and five years in the Mathematics Department of Rutgers University – Newark Campus.

He has been affiliated with the Mathematics Department of Baruch College since 1986, and was appointed to the Doctoral Faculty in Mathematics of the CUNY Graduate Center in 2006. He was the doctoral advisor of Shoshana Friedman (Ph.D. CUNY 2009) and doctoral co-advisor of Grigor Sargsyan (Ph.D. UC Berkeley 2009), whom he mentored as an undergraduate in the CUNY Baccalaureate Program. He has also supervised two additional students in advanced reading courses in mathematics as undergraduates, Lilit Martirosyan and Chase Skipper.

CURRENT SCHOLARLY INTERESTS:

Professor Apter is a mathematical logician, who specializes in set theory. His research focuses on large cardinals and forcing, but he also maintains a keen interest in inner model theory.

# Bagaria’s characterization of bounded forcing axioms in terms of generic absoluteness

Goldstern and Shelah (1995) introduced the class of bounded forcing axioms, that is forcing axioms for families of antichains of bounded size. For example, the bounded proper forcing axiom ${\rm BPFA}$ asserts that for any proper forcing notion $\mathbb{P}$ and any collection $D$ of at most $\aleph_1$ many maximal antichains in $\mathbb{P}$, each of size at most $\aleph_1$, there is a filter on $\mathbb{P}$ meeting each antichain in $D$. The speaker will present a theorem of Joan Bagaria (2000) that characterizes bounded forcing axioms in terms of generic absoluteness: for instance, Bagaria’s result shows that ${\rm BPFA}$ is equivalent to the assertion that if a $\Sigma_1$ sentence of the language of set theory with parameters of hereditary size at most $\aleph_1$ is true in some proper forcing extension, then it is already true in the ground model.

# On forcing and the (elusive) free two-generated left distributive algebra

We begin with an extended introduction to free left distributive algebras (LDs) including a normal form theorem for the one-generated free LD, which itself arises naturally from the assumption of a very large cardinal axiom. After discussing some applications and open problems, we make remarks on the difficulty of using forcing to attempt to construct a two-generated free LD by lifting the rank-to-rank elementary embedding used to create the one-generated free LD.

This is joint work with Joel David Hamkins.

# Grounded Martin’s axiom

*I will present a weakening of Martin’s axiom which asserts the existence of partial generics only for ccc posets contained in a ccc ground model. This principle, named the grounded Martin’s axiom, emerges naturally in the analysis of the Solovay-Tennenbaum proof of the consistency of MA. While the grounded MA has some of the combinatorial consequences of MA, it will be shown to be more flexible (being consistent with a singular continuum, for example) and more robust under forcing (being preserved in a strong way under both adding a Cohen or a random real).*

# Killing Inaccessible Cardinals Softly

I shall introduce the killing-them-softly phenomenon among large cardinals by showing how it works for inaccessible and Mahlo cardinals. A large cardinal is killed softly whenever, by forcing, one of its large cardinal properties is destroyed while as many as possible weaker large cardinal properties, below this one, are preserved. I shall also explore the various degrees of inaccessibility and show Mahlo cardinals are $alpha$-hyper$^{beta}$-inaccessible and beyond.

# Some problems motivated by nonstandard set theory

Nonstandard set theory enriches the usual set theory by a unary “standardness” predicate. Investigations of its foundations raise a number of questions that can be formulated in ZFC or GB and appear open. I will discuss several such problems concerning elementary embeddings, ultraproducts, ultrafilters and large cardinals.

# Survey on the structure of the Tukey theory of ultrafilters

The Tukey order on ultrafilters is a weakening of the well-studied Rudin-Keisler order, and the exact relationship between them is a question of interest. In second vein, Isbell showed that there is a maximum Tukey type among ultrafilters and asked whether there are others. These two questions are the main guiding forces of the current research. In this talk, we present highlights of recent work of Blass, Dobrinen, Mijares, Milovich, Raghavan, Todorcevic, and Trujillo (in various combinations for various papers). Further information about results mentioned in this talk can be found in a recent survey article by the speaker.

# Canonical Ramsey theory on Polish spaces

I would like to give an overview of recent results in canonical Ramsey theory in the context of descriptive set theory. This is the subject of a recent monograph joint with with Vladimir Kanovei and Jindra Zapletal. The main question we address is the following. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? Canonical Ramsey theory stems from finite combinatorics and is concerned with finding canonical forms of equivalence relations on finite (or countable) sets. We obtain canonization results for analytic and Borel equivalence relations and in cases when canonization is impossible, we prove ergodicity theorems. For a publisher’s book description see:

# Embeddings among the $\omega_1$-like models of set theory, part II

The speaker will give the second part of her talk, continued from the previous week. An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.

# Embeddings among the $\omega_1$-like models of set theory, part I

An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.

# Prikry-type sequences, iterations and critical sequences

I will present some old, some new and some classic results on the kinds of sequences which are added by some forcings which are related to Prikry forcing, in some sense. After finding combinatorial properties characterizing these sequences, I will show how to iterate the universe in such a way that the critical sequence of that iteration will satisfy that combinatorial property with respect to the target model, rendering it generic. This connection enables us to analyze precisely the collection of generic sequences which are present in one specific forcing extension. Time permitting, I will also explore connections to Boolean ultrapowers.

# Generic choice functions and ultrafilters on the integers

We will discuss a question asked by Stefan Geschke, whether the existence of a selector for the equivalence relation $E_0$ implies the existence of a nonprincipal ultrafilter on the integers. We will present a negative solution which is undoubtedly more complicated than necessary, using a variation of Woodin’s $mathbb{P}_{mathrm{max}}$. This proof shows that, under suitable hypotheses, if $E$ is a universally Baire equivalence relation on the reals, with countable classes, then forcing over $L(E,mathbb{R})$ to add a selector for $E$ does not add a nonprincipal ultrafilter on the integers.