Welcome to New York Logic!

Oct 23 Thursday | Oct 24 Friday | Oct 25 Saturday | Oct 26 Sunday | Oct 27 Monday | Oct 28 Tuesday | Oct 29 Wednesday |
---|---|---|---|---|---|---|

**→** go to the Calendar

## Upcoming talks and events:

# Morphic Modules and the Ziegler Spectrum

We will review the definitions of morphic rings and modules, and prove a result concerning conditions equivalent to total uniform morphicity of a ring (all R-modules being morphic in a uniform way). We consider questions such as, which abelian groups are morphic, and when products of morphic modules are morphic. This will lead us to a discussion of the Ziegler spectrum, and its relation to morphicness.

# Mutual Stationarity and Prikry-type forcings

Mutual stationarity is a property first introduced by Foreman and Magidor to study saturation properties of nonstationary ideals. Given a sequence $\langle\kappa_i : i < \lambda\rangle$ of regular cardinals, a sequence $\langle S_i: i < \lambda\rangle$ with $S_i \subseteq \kappa_i$ stationary for every $i$, is mutually stationary iff there are stationarily many subsets $A \subseteq \sup_{i < \lambda} \kappa_i$ s.t. $\sup(A \cap \kappa_i) \in S_i$ for all $i$ with $\kappa_i \in A$.

Consider this second property of a sequence $\langle\kappa_i : i < \lambda\rangle$: there is a forcing $P$ that changes $\text{cof}(\kappa_i)$ to $\eta_i$ without changing cofinalities or cardinalites of ordinals below $\inf{\kappa_i : i < \lambda}$.

We want to discuss how, and why, these properties are related.

# Set theory without the infinite

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.

# Independence in tame abstract elementary classes

Good frames are one of the main notions in Shelah’s classification theory for abstract elementary classes. Roughly speaking, a good frame describes a local forking-like notion for the class. In Shelah’s book, the theory of good frames is developed over hundreds of pages, and many results rely on GCH-like hypotheses and sophisticated combinatorial set theory.

In this talk, I will argue that dealing with good frames is much easier if one makes the global assumption of tameness (a locality condition introduced by Grossberg and VanDieren). I will outline a proof of the following result: Assume K is a tame abstract elementary class which has amalgamation, no maximal models, and is categorical in a cardinal of cofinality greater than the tameness cardinal. Then K is stable everywhere and has a good frame.

# Scott’s problem for models of ZFC

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.

# Interpretations and differential Galois extensions

We prove a number of results around finding strongly normal extensions of a differential field *K*, sometimes with prescribed properties, when the constants of *K* are not necessarily algebraically closed. The general yoga of interpretations and definable groupoids is used (in place of the Tannakian formalism in the linear case).

This is joint work with M. Kamensky.

# Mordell-Lang and Manin-Mumford in positive characteristic, revisited

We give a reduction of function field Mordell-Lang to function field Manin-Mumford, in positive characteristic. The upshot is another account of or proof of function field Mordell-Lang in positive characteristic, avoiding the recourse to difficult results on Zariski geometries.

(This work is joint with Benoist and Bouscaren.)

# The Rise and Fall of Accuracy-first Epistemology

Accuracy-first epistemology aims to supply non-pragmatic justifications for a variety of epistemic norms. The contemporary roots of accuracy-first epistemology are found in Jim Joyce’s program to reinterpret de Finetti’s scoring-rule arguments in terms of a “purely epistemic” notion of “gradational accuracy.” On Joyce’s account, scoring rules are conceived to measure the accuracy of an agent’s belief state with respect to the true state of the world, and Joyce argues that this notion of accuracy is a purely epistemic good. Joyce’s non-pragmatic vindication of probabilism, then, is an argument to the effect that a measure of gradational accuracy so imagined satisfies conditions that are close enough to those necessary to run a de Finetti style coherence argument. A number of philosophers, including Hannes Leitgeb and Richard Pettigrew, have joined Joyce’s program and gone whole hog. Leitgeb and Pettigrew, for instance, have argued that Joyce’s arguments are too lax and have put forward conditions that narrowing down the class of admissible gradational accuracy functions, while Pettigrew and his collaborators have extended the list of epistemic norms receiving an accuracy-first treatment, a program that he calls Evidential Decision Theory.

In this talk I report on joint work with Conor Mayo-Wilson that aims to challenge the core assumption of Evidential Decision Theory, which is the very idea of supplying a truly non-pragmatic justification for anything resembling the Von Neumann and Morgenstern axioms for a numerical epistemic utility function. Indeed, we argue that none of axioms have a satisfactory non-pragmatic justification, and we point to reasons why to suspect that not all the axioms could be given a satisfactory non-pragmatic justification. Our argument, if sound, has ramifications for recent discussions of “pragmatic encroachment”, too. For if pragmatic encroachment is a debate to do with whether there is a pragmatic component to the justification condition of knowledge, our arguments may be viewed to attack the true belief condition of (fallibilist) accounts of knowledge.

# Representing Scott sets in algebraic settings

The long standing problem of representing Scott sets as standard systems of models of Peano Arithmetic is one of the most vexing in the subject. We show that the analogous question has a positive solution for real closed fields and Presburger arithmetic. This is joint work with Alf Dolich, Julia Knight and Karen Lange.

# The general solution of a first order differential polynomial

This is the title of a 1976 paper by Richard Cohn in which he gives a purely algebraic proof of a theorem (proved analytically by Ritt) that gives a bound on the number of derivatives needed to find a basis for the radical ideal of the general solution of such a polynomial. I will show that the method introduced by Cohn can be used to give a modern proof of a theorem of Hamburger stating that a singular solution of such a polynomial is either an envelope of a set of solutions or embedded in an analytic family of solutions depending on whether or not it corresponds to an essential singular component of this polynomial. I will also discuss the relation between this phenomenon and the Low Power Theorem. This will be an elementary talk with all these terms and concepts defined and explained.

# Does definiteness-of-truth follow from definiteness-of-objects?

This talk — a mix of mathematics and philosophy — concerns the extent to which we may infer definiteness of truth in a mathematical context from definiteness of the underlying objects and structure of that context. The philosophical analysis is based in part on the mathematical observation that the satisfaction relation for model-theoretic truth is less absolute than often supposed. Specifically, two models of set theory can have the same natural numbers and the same structure of arithmetic in common, yet disagree about whether a particular arithmetic sentence is true in that structure. In other words, two models can have the same arithmetic objects and the same formulas and sentences in the language of arithmetic, yet disagree on their corresponding theories of truth for those objects. Similarly, two models of set theory can have the same natural numbers, the same arithmetic structure, and the same arithmetic truth, yet disagree on their truths-about-truth, and so on at any desired level of the iterated truth-predicate hierarchy. These mathematical observations, for which I shall strive to give a very gentle proof in the talk (using only elementary classical methods), suggest that a philosophical commitment to the determinate nature of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure N = {0,1,2,…} itself, but rather seems to be an additional higher-order commitment requiring its own analysis and justification.

This work is based on my recent paper, Satisfaction is not absolute, joint with Ruizhi Yang (Fudan University, Shanghai).

# Some new maximum VC classes

Vapnik-Chervonenkis classes with the maximum property are in some sense the most perfect set systems of finite Vapnik-Chervonenkis dimension. Definability of maximum VC classes in model theoretic structures is closely tied to other measures of complexity such as dp-rank. In this talk we show that set systems realizable as sets of positivity for linear combinations of real analytic functions have the maximum property on sets in general position. This may have applications to proving lower bounds on dp-rank in certain theories.

# Title TBA

# Countable Model Theory and the Complexity of Isomorphism

We discuss the Borel complexity of the isomorphism relation (for countable models of a first order theory) as the “right” generalization of the model counting problem. In this light we present recent results of Dave Sahota and the speaker which completely characterize the complexity of isomorphism for o-minimal theories, as well as recent work of Laskowski and Shelah which give a partial answer for omega-stable theories. Along the way, we introduce a few open problems and barriers to generalizing the existing results.