Friday / 9 May 2025 / 5pm
Speaker: Eli Gadsby
Title:
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Location: Thesis room
Friday / 2 May 2025 / 5pm
Speaker: Reilly
Title:
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Location: Thesis room
Friday / 25 April 2025 / 5pm
Speaker: Jason Block
Title:
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Location: Thesis room
Friday / 28 March 2025 / 5pm
Speaker: Valera Sergeev
Title: Geometric tools for decidability
Abstract: We will discuss some generalizations of Hilbert's Tenth Problem, and how Algebraic Geometry can be used to solve some of them. The main focus will be on the version of the Hilbert's Tenth Problem for the ring of formal power series over a finite field.
Location: Thesis room
Friday / 21 March 2025 / 5pm
Speaker: Davide Sutto (Oslo)
Title: Axiomatizing the Weak Iterative Conception of Set
Abstract: The iterative conception of set has formed centre-stage for the discussion of the philosophy of sets: sets are built in layers through the iteration of a process that generates new sets from old ones. The “typical” way of thinking of this conception is to generate all possible subsets when moving from one layer to another (i.e., generate sets through power-set). Call this the strong iterative conception of set. On the contrary, call the conception which takes no stand on whether we generate all subsets at additional layers the weak iterative conception of set. The idea can be traced back to Gödel’s “constructible universe” where one iterates definable (instead of full) power-set. Despite this distinguished representative, the weak conception is less studied, in terms of an axiomatic level-theoretic treatment, than the strong sharpening. This work seeks to remedy that deficit by proposing a Constructible Level Theory (CLT) which also sheds new light on the origins of the constructible universe as rooted in predicativist ideas. This is showcased by a choice concerning the Axiom of Separation. In one case CLT is made equivalent to ZF + V = L, while, in another instance, it can derive KP (or KPω) + V = L.
This Is joint work with Neil Barton (Singapore).
Location: Thesis room
Friday / 7 March 2025 / 5pm
Speaker: Elijah Gadsby
Title: Slow Consistency
Abstract: An empirical phenomenon in logic is that “natural” theories — those of independent interest — can be well-ordered in terms of their logical strength (under varying notions of strength), with the stronger theories proving the consistency of the weaker ones. However, using self reference, one can manufacture theories that are deductively intermediate between a theory T and its “natural” strengthening T+Con(T). The question thus becomes: are there any such intermediate theories that are “natural” or “interesting.”
A (plausible) example of such a theory intermediate between PA and PA+Con(PA) was given by Rathjen, Friedman, and Weiermann based on the notion of slow consistency. In the talk, I will explain this notion and outline a novel proof that the resulting theory is indeed intermediate. Following this, I will discuss how the ideas can be extended to exhibit infinite ascending and descending hierarchies of intermediate theories.
Location: Thesis room
Friday / 28 February 2025 / 5pm
Speaker: Brandon Ward
Title: Kunen's Inconsistency Theorem
Abstract: In the previous meeting we discussed measurable cardinals and their characterization as the critical point of an elementary embedding from V into a transitive class M. Stronger large cardinals arise from demanding M resemble V more and more. Taking this to the limit one can ask: is it consistent with ZFC that there exists a nontrivial elementary embedding from V to V? Kunen's Inconsistency Theorem tells us the answer is no. We'll discuss the theorem and its proof.
Location: Thesis room
Friday / 7 February 2025 / 5pm
Speaker: Brandon Ward
Title: Measurable Cardinals
Abstract: Large cardinals serve to strengthen ZF(C). They often arise by considering a property of omega, then demanding an uncountable cardinal with that property. From another view, large cardinals can often be characterized in terms of the existence of an elementary embedding from the universe V to some class M; the more M resembles V, the stronger the large cardinal assumption. We'll look at one of the earliest large cardinal notions, that of a measurable cardinal, due to Ulam, and its elementary embedding characterization, due to Scott.
Location: Thesis room
Friday / 13 December 2024 / 5pm
Speaker: Brandon Ward
Title: The Independence of AC from ZF via HOD
Abstract: The Axiom of Choice is independent of the standard axioms of set theory ZF. This is two statements: if ZF is consistent, so is ZF+AC; and if ZF is consistent, so is ZF+not(AC). To prove both of these we will look to notion of ordinal definability. In any given model of ZF, the hereditarily ordinal definable sets HOD is a definable inner model of ZF+AC, thus establishing the first half of the independence. To prove the second half we look upwards. We force over a countable transitive model to add countably many Cohen reals, then pass to a definable intermediate model, the model N of all sets hereditarily ordinal definable from parameters in the transitive closure of {X}, where X is set of Cohen reals. In N, there is no enumeration of the set X, and hence choice fails, thereby establishing the theorem.
Location: Thesis room
Friday / 8 November 2024 / 5pm
Speaker: Jason Block
Title: Some Computability Results on Profinite Groups
Abstract:
Location: Thesis room
Friday / 25 October 2024 / 5pm
Speaker: Eli Gadsby
Title: Goodstein's Theorem
Abstract: Goodstein’s Theorem is a result stating that certain sequences of natural numbers always terminate and is noteworthy for being one of the first examples of a natural combinatorial statement unprovable in PA. While on the surface the statement seems odd, it is actually a very well-motivated theorem and its unprovability in PA was by design. In this talk, I will discuss a paper by Rathjen on the topic and cover Goodstein sequences, their connection to ordinals below \epsilon_0 and Gentzen’s consistency proof, and a proof that Goodstein’s Theorem is unprovable in PA. Along the way, we will discuss some of the history of near-independence results.
Location: Thesis room
Friday / 27 September 2024 / 5 PM
Speaker: Emma Dinowitz
Title: Geometric Measure Theory
Abstract: In this talk, we will introduce Hausdorff measure and Hausdorff dimension, followed by their computable counterpart, effective Hausdorff dimension. We will then discuss the fundamental concepts in geometric measure theory of rectifiable and purely unrectifiable sets. The main focus of our presentation will be on the effectivization of these notions from a computability perspective. We will prove several theorems about the relation between the spectrum of A-computable variants of these notions and the standard notion.
Location: Thesis room
5/13/2024 3:00 PM
Speaker: Ben Goodman
Title: Set-Theoretic Geology
Abstract: Set-theoretic geology is a relatively new subfield, pioneered in large part by CUNY-affiliated set theorists. Whereas the method of forcing creates new, larger models by expanding the universe of sets, set-theoretic geology reverses this by exploring how the universe could have itself arisen by forcing over some smaller model. In this talk, I will survey the earlier results which the topic grew out of, the foundational paper of Fuchs, Hamkins, and Reitz, and the subsequent breakthroughs by Usuba.
Location: Room 3307
5/6/2024 3:00 PM
Speaker: Brandon Ward
Title: Lawvere's fixed-point theorem and diagonal arguments (pt. 3)
Abstract: Today we'll cover the background category theory to understand, state, and prove Lawvere's theorem in its full generality.
Location: Room 3307
4/1/2024 3:00 PM
Speaker: Brandon Ward
Title: Lawvere's fixed-point theorem and diagonal arguments (pt. 2)
Abstract: In part 2, I will go over some examples of Lawvere's theorem we did not cover in part 1.
Location: Room 3307
2/26/2024 3:00 PM
Speaker: Brandon Ward
Title: Lawvere's fixed-point theorem and diagonal arguments
Abstract: Guided by Lawvere’s original paper “Diagonal arguments and Cartesian closed categories” and Yanofsky’s exposition thereof, I will introduce and prove Lawvere’s fixed-point theorem. From its proof we will extract a simple scheme that underlies many self-referential paradoxes, incompleteness theorems, and fixed-point theorems.
Location: Room 3307
10/30/2023 1:45 PM (Early start to accommodate the speaker's schedule)
Speaker: Emma Dinowitz
Title: Computability of Equilibrium Measures
Abstract: We prove the computability of equilibrium measures for Hölder continuous potentials on a shift of finite type, with the Hölder constant bounded above. Our method is to use a computable version of the Ruelle-Perron-Frobenius theorem.
Location: Room 6421
10/16/2023 2:00 PM
Speaker: Ben Goodman
Title: Fixed Points of Functions on the Ordinals
Abstract: After surveying various natural situations in which fixed points of functions from the ordinals to the ordinals arise, I will show how to generalize such arguments and produce a large class of fixed points of any unbounded continuous nonstrictly-order-preserving function. Along the way we'll survey some basic topics in ordinal combinatorics.
Location: Room 6421
Postponed indefinitely
Speaker: Ben Goodman
Title: Accounts of Forcing and the Boolean Ultrapower
Abstract: Intuitively, the technique of forcing proceeds by enlarging the set-theoretic universe V with a generic object G to produce a larger universe V[G] with desired properties. Taken literally, this is nonsense, since V by definition already contains all sets and cannot be enlarged. In this talk, I will outline the three major approaches set theorists have taken to avoid this issue and make formal sense of forcing. First, there is the syntactic approach, where we simply define a method of coherently assigning truth values to sentences in an extended language and regard any notion of this corresponding to the truths of a larger universe as purely metaphorical. Second, there is the countable transitive model method, where instead of working over V we work over some countable model M; it is then easy to find objects generic over M, but additional metamathematical difficulties arise. Finally, there is the so-called "naturalist account of forcing" championed by Joel David Hamkins, where we replace V with its Boolean ultrapower, a (possibly ill-founded) class model into which it elementarily embeds and which is contained in an even larger model which can be thought of as a forcing extension of it.
Location: Room 6300
4/25/2023 2:00 PM
Speaker: Jason Block
Title: Computable Fields and Transcendence Bases
Abstract: [The speaker will determine which aspects of the topic to talk about based on the audience present on the day of the talk.]
Location: Room 6300
4/18/2023
Talk postponed
4/11/2023
Spring break, no meeting
4/4/2023 2:00 PM
Meeting cancelled
3/28/2023 2:00 PM
Speaker: Emma Dinowitz
Title: Boolean Valued Models, Part 3
Abstract:
Location: Room 6300
3/21/2023 2:00 PM
Speaker: Brandon Ward
Title: The Structure of the Paradoxes of Self-Reference
Abstract: I’ll present the paper of the same title by Graham Priest (1994). If you’ve encountered paradoxes of self reference, you might have got the feeling that there’s some underlying principle common to them all. Yet identifying this commonality proves difficult. Russell believed they are all of the same type, but post Ramsey, the orthodoxy was that there are two distinct families. The conclusion of this talk is that Ramsey was wrong and Russell was right. I’ll present some of the paradoxes of self-reference, then show how most of them fit Russell’s schema; the others will lead us to define the Qualified Russell’s schema. Assuming the axiom of choice, I’ll show that these two are equivalent. I’ll finish by discussing what to make of this result as it relates to the solutions of paradoxes in general.
Location: Room 6300
3/14/2023 2:00 PM
Speaker: Emma Dinowitz
Title: Boolean Valued Models, Part 2
Abstract: This is the second in a sequence of talks on boolean valued models of ZFC and forcing. In this talk we will discuss the mixing lemma, the maximum principle, and prove (some of) the axioms of ZFC in a boolean valued model.
Location: Room 6300
3/7/2023 2:00 PM
Speaker: Ben Goodman
Title: Large Cardinals Characterized by Elementary Embeddings on the Universe
Abstract: Measurable cardinals were originally defined in terms of highly complete nonprincipal ultrafilters (or equivalently highly additive binary-valued measures), but via the ultrapower construction they can be recast in terms of the existence of elementary embeddings of the set-theoretic universe V into some inner model M. I will outline this construction, then explore the stronger large cardinal notions that arise from imposing additional closure conditions on M, including the strong, superstrong, supercompact, and huge cardinals. If time allows, I will also discuss Kunen's famous theorem that there are no nontrivial elementary embeddings from V to V, as well as the large cardinal axioms that arise from skirting the edges of what Kunen ruled out.
Location: Room 6300
2/28/2023 2:00 PM
Speaker: Davide Leonessi
Title: Infinite Games
Abstract:
Location: Room 6300
2/21/2023 2:00 PM
Speaker: Emma Dinowitz
Title: Introduction to Boolean-Valued Models
Abstract: In this talk we will define Boolean valued models of ZFC, define the truth valuation on a Boolean valued model, and prove the truth of the axioms of ZFC in Boolean valued models. This is the first in a planned series of talks by Emma Dinowitz and Ben Goodman on forcing, Boolean-valued models, and non-classical logics.
Location: Room 6300
2/14/2023
No meeting
2/7/2023 2:00 PM
Speaker: Ben Goodman
Title: Meet the Large Cardinals
Abstract: After a brief introduction to the history and motivation of the large cardinal hierarchy, I will elaborate on specific large cardinal notions as directed by audience interest.
Location: Room 6300
11/15/2022 12:45 PM
Speaker: Ben Goodman
Title: Paradoxical Partitions of the Reals in ZF + "all sets are measurable"
Abstract: The controversial axiom of choice famously has many counterintuitive consequences, often related to the existence of non-measurable sets, which violate conventional notions of volume and probability. This has led to interest in alternative set theories without choice where every set of reals is measurable. However, these theories turn out to have an arguably even more counterintuitive implication: that there is a collection of non-empty disjoint sets of reals such that the cardinality of the collection is strictly greater than the cardinality of the reals.
11/8/2022 1:00 PM (note later start)
Speaker: Brandon Ward
Title: Godel's Constructible Universe
Abstract: In 1938, Gödel defined the constructible universe L within a model of ZF. He showed that L is a model of ZF satisfying the axiom of choice (AC) and the generalized continuum hypothesis (GCH), thereby proving the relative consistency of ZF with AC+GCH. In this talk we cover the recursive construction and prove the results. Without assuming AC in the ambient universe, we well-order L, thus showing AC holds there. Then confining ourselves to L, we show the sets with transitive closure smaller than k >= omega are precisely the constructible sets up to stage k, which yields GCH as a corollary.
Location: Room 3212
10/25/2022
No meeting
10/18/2022 12:30 PM
Speaker: Ben Goodman
Title: Arrow's Impossibility Theorem via Ultrafilters
Abstract: Arrow's impossibility theorem is a foundational result in social choice theory which shows that there is no voting system which a group of people can use to choose between more than two options which satisfies some seemingly reasonable criteria. One can recast this result in terms more familiar to logicians by showing that a voting system satisfying Arrow's criteria is equivalent to a nonprincipal ultrafilter on the set of voters. Arrow's theorem then follows immediately from the observation that every ultrafilter on a finite set is principal; however, it is equally apparent that if we allow the set of voters to be infinite and assume (a weak form of) the axiom of choice, Arrow's theorem fails.
10/11/2022 12:45 PM
Speaker: Emma Dinowitz
Title: Weihrauch Reducibility
11/9/2021 6:30 PM
Speaker: Sergei Artemov
Title: The Art of Formalization and The Provability of Consistency.
Abstract: We offer a mathematical proof of consistency for Peano Arithmetic PA and demonstrate that this proof is formalizable in PA. This refutes the widespread belief that there exists no consistency proof of a system that can be formalized in the system itself. Gödel's Second Incompleteness theorem yields that PA cannot derive the consistency formula Con(PA). This does not interfere with our formalized proof of PA-consistency which is not a derivation of the consistency formula Con(PA).
Location: Zoom. Email jblock@gradcenter.cuny.edu for a link to the meeting.
11/2/2021 6:30 PM
Topic: Nonstandard Analysis
Speaker: Emma Dinowitz
Location: Zoom. Email jblock@gradcenter.cuny.edu for a link to the meeting.
10/19/2021 6:30 PM
Topic: Inaccessible Cardinals and Their Relatives
Speaker: Ben Goodman
Location: Zoom. Email jblock@gradcenter.cuny.edu for a link to the meeting.
10/5/2021 6:30 PM
Topic: Random Graphs
Speaker: Valeriy Sergeev
Location: Zoom. Email jblock@gradcenter.cuny.edu for a link to the meeting.
9/28/2021 6:30 PM
Topic: Computability Theory Introduction
Speaker: Jason Block
Location: Zoom. Email jblock@gradcenter.cuny.edu for a link to the meeting.
9/21/2021 6:30 PM
Topic: Set Theory Introduction
Speaker: Ben Goodman
Location: Zoom. Email jblock@gradcenter.cuny.edu for a link to the meeting.
9/14/2021 6:30 PM
Topic: Semester Introduction/Planning
Speaker:
Location: Zoom. Email jblock@gradcenter.cuny.edu for a link to the meeting.