# Blog Archives

# Topic Archive: large cardinals

# Large cardinals, AECs and category theory

Shelah’s Categoricity Conjecture is a central test question in the study of Abstract Elementary Classes (AECs) in model theory. Recently Boney has shown that under the assumption that sufficiently large strongly compact cardinals exist, the Shelah Categoricity Conjecture holds at successor cardinals. Lieberman and Rosicky have subsequently shown that AECs can be characterised in a very natural way in a category-theoretic setting, and with this perspective Boney’s result can actually be seen as a corollary of an old category-theoretic result of Makkai and Pare. Rosicky and I have now been able to improve upon this result of Makkai and Pare (and consequently Boney’s Theorem), obtaining it from α-strongly compact cardinals.

# Namba-like Forcings at Successors of Singular Cardinals

Following up on Peter Koepke’s Logic Workshop lecture of March 22, 2013, I will discuss Namba-like forcings which either exist or can be forced to exist at successors of singular cardinals.

# A natural strengthening of Kelley-Morse set theory

I shall introduce a natural strengthening of Kelley-Morse set theory KM to the theory we denote KM+, by including a certain class collection principle, which holds in all the natural models usually provided for KM, but which is not actually provable, we show, in KM alone. The absence of the class collection principle in KM reveals what can be seen as a fundamental weakness of this classical theory, showing it to be less robust than might have been supposed. For example, KM proves neither the Łoś theorem nor the Gaifman lemma for (internal) ultrapowers of the universe, and furthermore KM is not necessarily preserved, we show, by such ultrapowers. Nevertheless, these weaknesses are corrected by strengthening it to the theory KM+. The talk will include a general elementary introduction to the various second-order set theories, such as Gödel-Bernays set theory and Kelley-Morse set theory, including a proof of the fact that KM implies Con(ZFC). This is joint work with Victoria Gitman and Thomas Johnstone.

# Reflecting I_0

We will present an argument for reflecting the large cardinal axiom I_0 from marginally stronger large cardinals. This will involve presenting some of the theory of inverse limits, which R. Laver first studied in the context of reflecting large cardinals at this level. Along the way we will see many local reflection results below I_0 and state a strong form of reflection which is useful in other contexts.

# Generalized descriptive set theory with very large cardinals

We will discuss the structure *L(V _{λ+1})* and attempts to generalize facts of descriptive set theory to this structure in the presence of very large cardinals. In particular we will introduce an axiom called Inverse Limit Reflection which we will argue is analogous to the Axiom of Determinacy in this context. The slides for this talk are available here.

# 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.

# Large cardinals need not be large in HOD

I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I shall begin with the example of a measurable cardinal that is not measurable in HOD, and then afterward describe how to force more extreme examples, such as a model with a supercompact cardinal, which is not weakly compact in HOD. This is very recent joint work with Cheng Yong.

# Ramsey cardinals and the continuum function

In his famous theorem, Easton used the Easton Product forcing to show that if $V\models{\rm GCH}$ and $F$ is any weakly increasing function on the regular cardinals such that $\text{cf}(F(\alpha))>\alpha$, then there is a cofinality preserving forcing extension in which $F$ is realized as the continuum function. The investigation then shifted to identifying which continuum patterns are compatible with large cardinals. It is not difficult to see that large cardinals affect the behavior of the continuum function. Obviously, if $\kappa$ is inaccessible, then by definition, the continuum function must have a closure at $\kappa$. Some other large cardinal influences are much more subtle. Easton’s original forcing does not work well in the presence of large cardinals; it, for instance, destroy weak compactness over $L$. So set theorists have had to develop some general and other very specific forcing techniques to address the behavior of the continuum function for a given large cardinal. In this talk, we will show that if $V\models{\rm GCH}$, $\kappa$ is Ramsey, and $F$ is any weakly increasing class function on the regular cardinals with a closure point at $\kappa$ such that $\text{cf}(F(\alpha))>\alpha$, then there is a cofinality preserving forcing extension in which $\kappa$ remains Ramsey and $F$ is realized as the continuum function. This is joint work with Brent Cody.

An extended abstract can be found here.

# 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.

# 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.

# Indestructibility for Ramsey cardinals

A large cardinal $\kappa$ is said to be indestructible by a certain poset $\mathbb P$ if $\kappa$ retains the large cardinal property in all forcing extensions by $\mathbb P$. Since most relative consistency results for ${\rm ZFC}$ are obtained via forcing, the knowledge of a large cardinal’s indestructibility properties is used to establish the consistency of that large cardinal with other set theoretic properties. In this talk, I will use an elementary embeddings characterization of Ramsey cardinals to prove some basic indestructibility results.