# CUNY Logic Workshop

The CUNY Logic Workshop is a weekly seminar in mathematical logic at the CUNY Graduate Center.
(50 items)

CUNY Logic WorkshopFriday, December 13, 20132:00 pmGC 6417

# Higher-order Reverse Mathematics: Where existence meets computation via infinitesimals.

Department of Mathematics, Ghent University

Classically, the existence of an object tells us very little about how to construct said object.
We consider a nonstandard version of Ulrich Kohlenbach’s higher-order Reverse Mathematics
in which there is a very elegant and direct correspondence between, on one hand, the existence
of a functional computing an object and, on the other hand, the classical existence of this object
with the same standard and nonstandard properties. We discuss how these results -potentially-
contribute to the programs of finitistic and predicativist mathematics.

CUNY Logic WorkshopFriday, December 6, 20132:00 pmGC 6417

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

City Tech - CUNY

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.

CUNY Logic WorkshopFriday, November 22, 20132:00 pmGC 6417

# Measure semantics for modal logics

Columbia University

Long before Kripke semantics became standard in modal logic, Tarski showed us that the basic propositional modal language can be interpreted in topological spaces. In Tarski’s semantics for the modal logic $S4$, each propositional variable is evaluated to an arbitrary subset of a fixed topological space. I develop a related, measure theoretic semantics, in which modal formulas are interpreted in the Lebesgue measure algebra, or algebra of Borel subsets of the real interval $[0,1]$, modulo sets of measure zero. This semantics was introduced by Dana Scott in the last several years. I discuss some of my own completeness results, and ways of extending the semantics to more complex modal languages.

CUNY Logic WorkshopFriday, November 15, 20132:00 pmGC 6417

# The recursive spectrum of a strongly minimal modular theory of groups in a finite language

City College -- CUNY
CUNY Logic WorkshopFriday, November 8, 20132:00 pmGC 6417

# Prikry-type sequences, iterations and critical sequences

The City University of New York

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.

CUNY Logic WorkshopFriday, November 1, 20132:00 pmGC 6417

# Reverse VC Calculations

Haverford College

Note Hill’s Workshop talk is cancelled due to illness.

Let F be a family of sets, for example, a uniformly definable semi-algebraic family in real or p-adic n-space. The Vapnik-Chervonenkis (VC) dimension of F is a measurement of the combinatorial complexity of F. Once you know the VC dimension of F, theorems from computational geometry, like the Epsilon-Net Theorem, give nice geometric consequences for F. I will discuss a statistical strategy for reversing the flow of information in this theorem. Instead of starting with knowledge of the VC dimension, we merely hypothesize “dimension=d” for some value d. Then, we observe the geometric behavior of F using computer experiments and compare the observed behavior with the behavior that is predicted by the theorem (under the hypothesis “dimension=d”). If our observed results have sufficiently low probability (conditioned on “dimension=d”), then we can reject the hypothesis “dimension=d” with a high degree of confidence. Ultimately, we hope to use such methods to shed light on conjectures about VC density in the p-adics. This project is joint work with Deirdre Haskell and Nigel Pynn-Coates.

CUNY Logic WorkshopFriday, October 25, 20132:00 pmGC 6417

# Automorphism Groups of Countable, Recursively Saturated Models of Peano Arithmetic

University of Connecticut

It is still unknown whether there are nonisomorphic countable recursively saturated models M and N whose automorphism groups Aut(M) and Aut(N) are isomorphic. I will discuss what has happened over the last 20 years towards showing that such models do not exist, including some very recent results.

CUNY Logic WorkshopFriday, October 18, 20132:00 pmGC 6417

# Generic choice functions and ultrafilters on the integers

Miami University of Ohio

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.

CUNY Logic WorkshopFriday, October 11, 20132:00 pmGC 6417

# Randomness and ergodic theorems under measure-preserving transformations

University of Connecticut

Ergodic theorems describe regular measure-theoretic behavior, and a point is said to be algorithmically random if it has no rare measure-theoretic properties of a certain kind. The connections between these two types of regularity have been well studied over the past five years. I will discuss Birkhoff’s ergodic theorem with respect to transformations that are measure-preserving but not necessarily ergodic in the context of a computable probability space. Then I will show that each point in such a space that is not Martin-L”of random does not satisfy Birkhoff’s ergodic theorem with respect to every computable set and measure-preserving transformation.

This work is joint with Henry Towsner.

CUNY Logic WorkshopFriday, October 4, 20132:00 pmGC 6417

# Why model-theorists shouldn’t think that ACF is easy

The City University of New York

We all learned that stability theory derived many of its ideas from what happens in ACF, where everything is nice and easy. After all ACF has quantifier elimination and is strongly minimal, decidable, superstable, uncountably categorical, etc. However, my own struggles with ACF have humbled my opinion about it: it is an awfully rich theory that encodes way more than our current knowledge. I will discuss some examples showing how “difficult” ACF is: Grothendieck ring, isomorphism problem, set-theoretic intersection problem. Oddly enough, RCF seems to not have any of these problems. It is perhaps my ignorance, but I have come to think of RCF as much easier. Well, all, of course, is a matter of taste.

CUNY Logic WorkshopFriday, September 27, 20132:00 pmGC 6417

# Satisfaction is not absolute

The City University of New York

I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory $M_1$ and $M_2$, for example, can agree on their natural numbers $langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_1}=langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_2}$, yet disagree on arithmetic truth: they have a sentence $sigma$ in the language of arithmetic that $M_1$ thinks is true in the natural numbers, yet $M_2$ thinks $negsigma$ there. Two models of set theory can agree on the natural numbers $mathbb{N}$ and on the reals $mathbb{R}$, yet disagree on projective truth. Two models of set theory can have the same natural numbers and have a computable linear order in common, yet disagree about whether this order is well-ordered. Two models of set theory can have a transitive rank initial segment $V_delta$ in common, yet disagree about whether this $V_delta$ is a model of ZFC. The theorems are proved with elementary classical methods.

This is joint work with Ruizhi Yang (Fudan University, Shanghai). We argue, on the basis of these mathematical results, that the definiteness of truth in a structure, such as with arithmetic truth in the standard model of arithmetic, cannot arise solely from the definiteness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

CUNY Logic WorkshopFriday, September 20, 20132:00 pm

# A survey of the model theory of tracial von Neumann algebras

University of Illinois Chicago

Von Neumann algebras are certain algebras of bounded operators on Hilbert spaces. In this talk we will survey some of the model theoretic results about (tracial) von Neumann algebras, focusing mainly on (in)stability, quantifier-complexity, and decidability. No prior knowledge of von Neumann algebras will be necessary. Some of the work presented is joint with Ilijas Farah, Bradd Hart, David Sherman, and Thomas Sinclair.

CUNY Logic WorkshopFriday, May 10, 20132:00 pmGC 6417

# What is the theory ZFC without power set?

The New York City College of Technology (CityTech), CUNY

The theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed — specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered — is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context.

For example, there are models of ZFC- in which a countable union of countable sets is not countable. There are models of ZFC- for which the Los ultrapower theorem fails, even for wellfounded ultrapowers on a measurable cardinal. Moreover, the theory ZFC- is not sufficient to establish that the union of Σn and Πn sets is closed under bounded quantification. Lastly, there are models of ZFC- for which the Gaifman theorem fails, in that there exists cofinal embeddings j:M–>N between ZFC- models that are Σ1-elementary, but not fully elementary.

Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using collection rather than replacement in the axiomatization above. This is joint work with Joel David Hamkins and Victoria Gitman, and it extends prior work of Andrzej Zarach.

arxiv preprint | post at jdh.hamkins.org | post on Victoria Gitman’s blog

CUNY Logic WorkshopFriday, May 3, 20132:00 pmGC 6417

# Models of Reverse Mathematics

University of Pennsylvania

We discuss two results relating ideas in Reverse Mathematics to the properties of models of first order arithmetic. The first shows that we can extend second order arithmetic by the existence of a non-principal ultrafilter — a third order property — while remaining conservative. The second result shows that we can extend models of RCA so that any particular set is definable; this allows us to recover some properties of models of Peano arithmetic for models of RCA.

CUNY Logic WorkshopFriday, April 26, 20132:00 pmGC 6417

# Mittag-Leffler objects in definable categories of modules

The City University of New York

Definable categories are classes of modules closed under direct product, direct limit, and pure submodule. These play an important role in the theory of modules as they are in bijection with the closed sets of the Ziegler spectrum (and thus with the product closed complete theories of modules). Mittag-Leffler objects for such categories (some sort of generalized atomic module) will be introduced and a necessary and sufficient condition in terms of generators and generalized relations will be given for them to exist.

CUNY Logic WorkshopFriday, April 12, 20132:00 pmGC 6417

# An algebraic characterization of recursively saturated real closed fields

Wellesley College

We (with D’Aquino and Kuhlmann) give a valuation theoretic characterization for a real closed field to be recursively saturated. Previously, Kuhlmann, Kuhlmann, Marshall, and Zekavat gave such a characterization for kappa-saturation, for all infinite cardinals kappa. Our result extends the characterization for a divisible ordered abelian group to be recursively saturated found in some unpublished work of Harnik and Ressayre.

Friday, April 12, 201312:30 pmGC 6417

# Pfaffian functions vs. Rolle leaves

McMaster University

In the early 1980s, after Khovanskii’s ICM lecture, van den Dries formulated the conjecture that the expansion P of the real field by all pfaffian functions was model complete. Thinking about the problem led him to formulate a minimality notion in expansions of the real order, which directly inspired Pillay and Steinhorn in their discovery of o-minimality. However, while P has been known to be o-minimal since Wilkie’s groundbreaking work in 1996, van den Dries’s conjecture is still open today. Recently, Lion and I proved a variant of this conjecture, in which “pfaffian functions” are replaced with “nested Rolle leaves”, which in essence correspond to the objects originally studied by Khovanskii. The mystery lies in how these two expansions are related. I will explain each of them and exhibit a third related notion, found recently in joint work with Jones, which might clarify this relationship.

CUNY Logic WorkshopFriday, March 22, 20132:00 pmGC 6417

# Namba-like singularizations of successor cardinals

Rheinische Friedrich-Wilhelms-Universität Bonn

Bukowski-Namba forcing preserves aleph_1 and changes the cofinality of  aleph_2 to omega. We lift this to cardinals kappa > aleph_1 : Assuming a measurable cardinal lambda we construct models over which there is a further “Namba-like” forcing which preserves all cardinals <= kappa and changes the cofinality of kappa^+ to omega. Cofinalities different from omega can also be achieved by starting from measurable cardinals of sufficiently strong Mitchell order. Using core model theory one can show that the respective measurable cardinals are also necessary. This is joint work with Dominik Adolf (Münster).

Slides

CUNY Logic WorkshopFriday, March 22, 20134:00 pmGC 6417Note special time

# Determinacy in analysis and beyond

University of Bristol

Recently Montalban and Shore derived precise limits to the amount of determinacy provable in second order arithmetic.  We review some of the results in this area and recent work on lifting this to a setting of ZF^- with a single measurable cardinal.

Slides

CUNY Logic WorkshopFriday, March 15, 20132:00 pm

# Some approaches to model theory for classes of finite structures

Vassar College

This talk surveys work on the development of model theory for classes of finite structures due primarily to Macpherson and the speaker, and to several of Macpherson’s students. The underlying theme of this work has been to bring to the model-theoretic study of classes of finite structures aspects of the model theory of infinite structures.

CUNY Logic WorkshopFriday, March 8, 201312:00 amGC 6417

# Reverse mathematics for second-order categoricity theorem

Mathematical Institute, Tohoku University

It is important in the foundations of mathematics that the natural number system is characterizable as a system of 0 and a successor function by second-order logic. In other words, the following Dedekind’s second-order categoricity theorem holds: every Peano system $(P,e,F)$ is isomorphic to the natural number system $(N,0,S)$. In this talk, I will investigate Dedekind’s theorem and other similar statements. We will first do reverse mathematics over $RCA_0$, and then weaken the base theory. This is a joint work with Stephen G. Simpson.

CUNY Logic WorkshopFriday, March 1, 20132:00 pmGC 6417

# Understanding genericity for cuts

Ghent University

In a nonstandard model of arithmetic, initial segments with no maximum elements are traditionally called cuts. It is known that even if we restrict our attention to cuts that are closed under a fixed family of functions (e.g., multiplication, the primitive recursive functions, or the Skolem functions), the properties of cuts can still vary greatly. I will talk about what genericity means amongst such great variety. This notion of genericity comes from a version of model theoretic forcing devised by Richard Kaye in his 2008 paper. Some ideas were already implicit in the work by Laurence Kirby and Jeff Paris on indicators in the 1970s.

CUNY Logic WorkshopFriday, February 22, 20132:00 pmGC 6417

# Very NIP Ordered Groups

The City University of New York

In recent years there has been renewed interest in theories without the independence property (NIP theories), a class of theories including all stable as well as all o-minimal theories.  In this talk we concentrate on theories, T, which expand that of divisible ordered Abelian groups (a natural situation to consider if one is motivated by the study of o-minimal theories) and consider the problem determining the consequences of assuming that T is NIP on the structure of definable sets in models of T.  One quickly realizes that given the great generality of the NIP assumption  in order to address this type of question one wants to consider much stronger variants of not having the independence property.  Thus we are led to the study of definable sets in models of  theories T expanding that of divisible ordered Abelian groups satisfying various very strong forms of the NIP condition such as finite dp-rank, convex orderability, and VC minimality.  In this talk I will survey results in this area and discuss many open problems.

CUNY Logic WorkshopFriday, February 15, 201312:00 amGC 6417

# On the axiom of constructibility and Maddy’s conception of restrictive theories

The City University of New York

This talk will be based on my paper, A multiverse perspective on the axiom of constructibility.

Set-theorists often argue against the axiom of constructibility V=L on the grounds that it is restrictive, that we have no reason to suppose that every set should be constructible and that it places an artificial limitation on set-theoretic possibility to suppose that every set is constructible.  Penelope Maddy, in her work on naturalism in mathematics, sought to explain this perspective by means of the MAXIMIZE principle, and further to give substance to the concept of what it means for a theory to be restrictive, as a purely formal property of the theory.

In this talk, I shall criticize Maddy’s specific proposal.  For example, it turns out that the fairly-interpreted-in relation on theories is not transitive, and similarly the maximizes-over and strongly-maximizes-over relations are not transitive.  Further, the theory ZFC + `there is a proper class of inaccessible cardinals’ is formally restrictive on Maddy’s proposal, although this is not what she had desired.

Ultimately, I argue that the $Vneq L$ via maximize position loses its force on a multiverse conception of set theory, in light of the classical facts that models of set theory can generally be extended to (taller) models of V=L.  In particular, every countable model of set theory is a transitive set inside a model of V=L.  I shall conclude the talk by explaining various senses in which V=L remains compatible with strength in set theory.

CUNY Logic WorkshopFriday, February 8, 201312:00 amGC 6417

# Model theory of satisfaction classes

The City University of New York

All countable recursively saturated models of Peano Arithmetic have nonstandard satisfaction classes. In fact, each such model has a great variety of nonstandard satisfaction classes. I will survey model theoretic techniques that can be applied to construct many different inductive satisfaction classes, and I will show how, in return, inductive satisfaction classes are used to prove important result about recursively saturated models of PA. I will also pose an open problem concerning a possible converse to Tarski’s undefinability of truth theorem.

Pdf slides for the talk are here: Satisfaction Classes

CUNY Logic WorkshopFriday, September 28, 201212:00 amGC 6417This is the usual place

# Transfinite linear algebra

The City University of New York
CUNY Logic WorkshopFriday, September 14, 201212:00 amGC 6417

# Boolean subalgebras of the computable atomless Boolean algebra

City University of New York

An abstract of this talk will be added.

CUNY Logic WorkshopFriday, September 7, 201212:00 amGC 6417

# Recent progress on the modal logic of forcing and grounds

The City University of New York

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and“true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC. Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”. In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.

CUNY Logic WorkshopFriday, May 18, 201212:00 amGC 6417

# The countable models of ZFC, up to isomorphism, are linearly pre-ordered by the submodel relation; indeed, every countable model of ZFC, including every transitive model, is isomorphic to a submodel of its own L

The City University of New York

This will be a talk on some extremely new work. The proof uses finitary digraph combinatorics, including the countable random digraph and higher analogues involving uncountable Fraisse limits, the surreal numbers and the hypnagogic digraph.

The story begins with Ressayre’s remarkable 1983 result that if $M$ is any nonstandard model of PA, with $langletext{HF}^M,{in^M}rangle$ the corresponding nonstandard hereditary finite sets of $M$, then for any consistent computably axiomatized theory $T$ in the language of set theory, with $Tsupset ZF$, there is a submodel $Nsubsetlangletext{HF}^M,{in^M}rangle$ such that $Nmodels T$. In particular, one may find models of ZFC or even ZFC + large cardinals as submodels of $text{HF}^M$, a land where everything is thought to be finite. Incredible! Ressayre’s proof uses partial saturation and resplendency to prove that one can find the submodel of the desired theory $T$.

My new theorem strengthens Ressayre’s theorem, while simplifying the proof, by removing the theory $T$. We need not assume $T$ is computable, and we don’t just get one model of $T$, but rather all models—the fact is that the nonstandard models of set theory are universal for all countable acyclic binary relations. So every model of set theory is a submodel of $langletext{HF}^M,{in^M}rangle$.

Theorem.(JDH) Every countable model of set theory is isomorphic to a submodel of any nonstandard model of finite set theory. Indeed, every nonstandard model of finite set theory is universal for all countable acyclic binary relations.

The proof involves the construction of what I call the countable random $mathbb{Q}$-graded digraph, a countable homogeneous acyclic digraph that is universal for all countable acyclic digraphs, and proving that it is realized as a submodel of the nonstandard model $langle M,in^Mrangle$. Having then realized a universal object as a submodel, it follows that every countable structure with an acyclic binary relation, including every countable model of ZFC, is realized as a submodel of $M$.

Theorem.(JDH) Every countable model $langle M,in^Mrangle$ of ZFC, including the transitive models, is isomorphic to a submodel of its own constructible universe $langle L^M,in^Mrangle$. In other words, there is an embedding $j:Mto L^M$ that is quantifier-free-elementary.

The proof is guided by the idea of finding a universal submodel inside $L^M$. The embedding $j$ is constructed completely externally to $M$.

Corollary.(JDH) The countable models of ZFC are linearly ordered and even well-ordered, up to isomorphism, by the submodel relation. Namely, any two countable models of ZFC with the same well-founded height are bi-embeddable as submodels of each other, and all models embed into any nonstandard model.

The work opens up numerous questions on the extent to which we may expect in ZFC that $V$ might be isomorphic to a subclass of $L$. To what extent can we expect to have or to refute embeddings $j:Vto L$, elementary for quantifier-free assertions?