Blog Archives

Topic Archive: set theory

Set theory seminarFriday, December 5, 201412:00 pmGC 6417

Andrew Brooke-Taylor

A nice model for cardinal characteristics at a supercompact κ

University of Bristol

In a throw-away comment in a relatively recent preprint, Garti and Shelah state that using the technique of Dzamonja and Shelah, one can start with a model of set theory containing a supercompact cardinal κ, and force to obtain a model in which κ remains supercompact, 2κ is large, but the ultrafilter number at κ is only κ+. I will present this construction, and with it further results from joint work with Vera Fischer and Diana Montoya pinning down many other generalized cardinal characteristics at κ in the resulting model.

Set theory seminarFriday, November 7, 201412:00 pm6417

Gunter Fuchs

News on the Solid Core

The City University of New York

Set-theoretic geology, a line of research jointly created by Hamkins, Reitz and myself, introduced some inner models which result from inverting forcing in some sense. For example, the mantle of a model of set theory V is the intersection of all inner models of which V is an extension by set-forcing. It was an initial, naive hope that one might arrive at a model that is in some sense canonical, but one of the main results on set-theoretic geology is that this is not so: every model of set theory V has a class forcing extension V[G] so that the mantle, as computed in V[G], is V. So quite literally, the mantle of a model of set theory can be anything.

In an attempt to arrive at a concept that fits in with the general spirit of set-theoretic geology, but that stands a chance of being canonical, I defined a set to be solid if it cannot be added to an inner model by set-forcing, and I termed the union of all solid sets the “solid core”.

I will present some results on the solid core which were obtained in recent joint work with Ralf Schindler, and which show that the solid core indeed is a canonical inner model, assuming large cardinals (more precisely, if there is an inner model with a Woodin cardinal), but that it is not as canonical as one might have hoped without that assumption.

Set theory seminarFriday, October 31, 201412:00 pmGC 6417

Kameryn Williams

Scott’s problem for models of ZFC

The CUNY Graduate Center

Scott’s problem asks which Scott sets can be represented as standard systems of models of Peano arithmetic. Knight and Nadel proved in 1982 that every Scott set of cardinality $\le \omega_1$ is the standard system of some model of PA. Little progress has been made since then on the problem. I will present a modification of these results to models of ZFC and show that every Scott set of cardinality $\le \omega_1$ is the standard system of some model of ZFC. This talk will constitute the speaker’s oral exam.

CUNY Logic WorkshopFriday, November 7, 20142:00 pmGC 6417

Stevo Todorcevic

Choice principles and Ramsey theory

University of Toronto

This talk will provide an overview of results of Ramsey theory that have close relationships with constructions of models of ZF that distinguish between various forms of the Axiom of Choice. Some open problems and directions for further research will also be discussed.

Stevo Todorcevic
University of Toronto
NERDS: New England Recursion & Definability SeminarSaturday, October 18, 201410:30 amAssumption College, Worcester, MA

The implicitly constructible universe

Dartmouth College

The implicitly constructible universe, IMP, defined by Hamkins and Leahy, is produced by iterating implicit definability through the ordinals. IMP is an inner model intermediate between L and HOD. We look at some consistency questions about the nature of IMP.

Dartmouth College
Marcia Groszek is a professor on the faculty of Dartmouth College.
CUNY Logic WorkshopFriday, September 19, 20142:00 pmGC 6417

Victoria Gitman

Choice schemes for Kelley-Morse set theory

The City University of New York

Kelley-Morse (${\rm KM}$) set theory is one of the standard axiomatic foundations for set theory with classes as well as sets. Its defining feature is a strong class existence principle which states that any collection defined by a second-order assertion is a class. Choice schemes are choice/collection axioms for classes. The full choice scheme states for every second-order assertion $\varphi$ that if for every set $x$, there is a class $X$ such that $\varphi(x,X)$, then there is a single class $Z$ collecting all the witnesses, so that $\varphi(x,Z_x)$ holds, where $Z_x$ is the slice on coordinate $x$ of $Z$. The full choice scheme can be weakened in a number of ways. For instance, the set-sized choice scheme allows only set many choices to be made and the $\Sigma^1_n$-choice scheme restricts the complexity of $\varphi$. Study of the choice schemes dates back to the work of Marek and Mostowski from the 1970s. They have recently found application in nonstandard set theory with infinitesimals and analysis of properties of class forcing extensions. We show that even the weakest fragment of the choice scheme, where $\omega$-many choices must be made for a first-order assertion, may fail in a model of ${\rm KM}$ and that it is possible for the set-sized choice scheme to hold, while the full choice scheme fails for a first-order assertion. We argue that the theory ${\rm KM^+}$ consisting of ${\rm KM}$ together with the full choice scheme is more robust than ${\rm KM}$ because it is able to prove the Łoś Theorem for second-order ultrapowers and the absorption of first-order quantifiers in second-order assertions. We show that both these properties can fail in a model of ${\rm KM}$: the second-order ultrapower of a ${\rm KM}$-model may not even be a model of ${\rm KM}$ and a second-order assertion of complexity $\Sigma^1_n$ with a set quantifier in front may fail to have complexity $\Sigma^1_n$. This is joint work with Joel David Hamkins and Thomas Johnstone.

CUNY Logic WorkshopFriday, October 24, 20142:00 pmGC 6417

Laurence Kirby

Set theory without the infinite

Baruch College - CUNY

The historical origins of set theory lay in the study of the infinite. Later came the universalist claim that set theory is a foundation for all of mathematics. We consider the consequences of accepting the universalism without accepting the infinite. We come to see finite sets as graphs or as processes, the result of their own coming into being. Basic methods of set construction give rise to arithmetics of sets with some surprising properties.

Laurence Kirby
Baruch College - CUNY
Laurie Kirby received his Ph.D. from Manchester University in 1977. After spells in Paris and Princeton, he joined Baruch College as a professor in 1982.
Set theory seminarFriday, September 5, 201410:00 amGC 6417

Jose Mijares

Local Ramsey theory: an abstract approach

University of Denver

This talk is about an abstract version of the notion of semi-selective co-ideal for subsets of a topological Ramsey space. This version is useful to characterize the corresponding generalization of the local Ramsey property in “topological terms”. We will also talk about forcing notions related to this abstract version of semi-selectivity, generalizing those related to Ellentuck’s space, and we will comment on some applications.

CUNY Logic WorkshopFriday, December 12, 20142:00 pmGC 6417

Andrew Brooke-Taylor

Large cardinals, AECs and category theory

University of Bristol

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.

Andrew Brooke-Taylor
University of Bristol
Andrew Brooke-Taylor is a set theorist, who applies large cardinal axioms and other tools and techniques from set theory to other areas of mathematics, particularly category theory and algebraic topology. He received his doctorate in 2007 from the Kurt Gödel Research Center and the Universität Wien, and has held postdoctoral positions at the University of Bristol and at Kobe University.
CUNY Logic WorkshopFriday, September 5, 20142:00 pmGC 6417

Jose Mijares

Topological Ramsey spaces and Fraïssé structures

University of Denver

There seems to be a natural relationship between topological Ramsey spaces and Fraïssé classes of finite structures. In fact, for some Fraïssé classes satisfying the Ramsey property, it is possible to define a topological Ramsey space such that the Fraïssé limit of the class is essentially an element of the space. We will talk about examples of this phenomenon, describe the general case to some extent, and comment about how this could be understood as a abstract tool to classify Fraïssé structures.

Jose Mijares
University of Denver
Goyo Mijares received his Ph.D. in 2007 from the Central University of Venezuela, and is now a postdoctoral scholar at the University of Denver. He studies set theory, Ramsey theory, topological dynamics, combinatorics, and functional analysis.
Tuesday, May 13, 20143:50 pm750 Schapiro CEPSR, Columbia University Morningside Campus

Descriptive Graph Combinatorics and Computability

Cal Tech
George Leibman
Bronx Community College, CUNY
George Leibman is a professor in the Mathematics and Computer Science department at Bronx Community College, CUNY. He received his doctorate from the CUNY Graduate Center in 2004, under the direction of Joel Hamkins, and he conducts research in set theory, with a particular interest in the modal logic of forcing.
CUNY Logic WorkshopFriday, May 2, 20142:00 pmGC 6417

Joel David Hamkins

A natural strengthening of Kelley-Morse set theory

The City University of New York

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.

CUNY Logic WorkshopFriday, March 7, 20142:00 pmGC 6417

Scott Cramer

Generalized descriptive set theory with very large cardinals

Rutgers University

We will discuss the structure L(Vλ+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.