Set Theory Talks

Dino Rossegger, UC Berkeley and TU Wien
The structural complexity of models of PA

The Scott rank of a countable structure is the least ordinal α such that all automorphism orbits of the structure are definable by infinitary Σα formulas. Montalbán showed that the Scott rank of a structure is a robust measure of the structural and computational complexity of a structure by showing that various different measures are equivalent. For example, a structure has Scott rank α if and only if it has a Πα+1 Scott sentence if and only if it is uniformly ΔΔ0α categorical if and only if all its automorphism orbits are Σα infinitary definable.

In this talk we present results on the Scott rank of non-standard models of Peano arithmetic. We show that non-standard models of PA have Scott rank at least ω, but, other than that, there are no limits to their complexity. Given a completion T of PA we give a reduction via bi-interpretability of the class of linear orders to the models of T. This allows us to exhibit models of T of Scott rank α for every ω≤α≤ω1. In particular, every completion of T has models of high Scott rank.

This is joint work with Antonio Montalbán.

Speaker:     Gershom Bazerman, Arista Networks.

Date and Time:     Wednesday May 4, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Classes of Closed Monoidal Functors which Admit Infinite Traversals.

Weak Indestructibility and Reflection

Assuming multiple of strong cardinals, there are lots of cardinals with small degrees of strength (i.e. κ that are κ+2-strong). We can calculate the consistency strength of these all cardinal's small degrees of strength being weakly indestructible using forcing and core model techniques in a way similar to Apter and Sargsyan's previous work. This yields some easy relations between indestructibility and Woodin cardinals, and also generalizes easily to supercompacts. I will give a proof sketches of these results.

Michał Godziszewski, University of Vienna
Modal Quantifiers, Potential Infinity, and Yablo sequences

When properly arithmetized, Yablo's paradox results in a set of formulas which (with local disquotation in the background) turns out to be consistent, but ω-inconsistent. Adding either uniform disquotation or the ω-rule results in inconsistency. Since the paradox involves an infinite sequence of sentences, one might think that it doesn't arise in finitary contexts. We study whether it does. It turns out that the issue depends on how the finitistic approach is formalized. On one of them, proposed by Marcin Mostowski, all the paradoxical sentences simply fail to hold. This happens at a price: the underlying finitistic arithmetic itself is ω-inconsistent. Finally, when studied in the context of a finitistic approach which preserves the truth of standard arithmetic, the paradox strikes back - it does so with double force, for now the inconsistency can be obtained without the use of uniform disquotation or the ω-rule. This is joint work with Rafał Urbaniak from the University of Gdańsk.

Speaker:     Alex Sorokin, Northeastern University.

Date and Time:     Wednesday April 27, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     The defect of a profunctor.

Abstract: In the mid 1960s Auslander introduced a notion of the defect of a finitely presented functor on a module category. In 2021 Martsinkovsky generalized it to arbitrary additive functors. In this talk I will show how to define a defect of any enriched functor with a codomain a cosmos. Under mild assumptions, the covariant (contravariant) defect functor turns out to be a left covariant (right contravariant) adjoint to the covariant (contravariant) Yoneda embedding. Both defects can be defined for any profunctor enriched in a cosmos V. They happen to be adjoints to the embeddings of V-Cat in V-Prof. Moreover, the Isbell duals of a profunctor are completely determined by the profunctor's covariant and contravariant defects. These results are based on applications of the Tensor-Hom-Cotensor adjunctions and the (co)end calculus.

Dino Rossegger, UC Berkeley and TU Wien
The structural complexity of models of PA

The Scott rank of a countable structure is the least ordinal α such that all automorphism orbits of the structure are definable by infinitary Σα formulas. Montalbán showed that the Scott rank of a structure is a robust measure of the structural and computational complexity of a structure by showing that various different measures are equivalent. For example, a structure has Scott rank α if and only if it has a Πα+1 Scott sentence if and only if it is uniformly ΔΔ0α categorical if and only if all its automorphism orbits are Σα infinitary definable.

In this talk we present results on the Scott rank of non-standard models of Peano arithmetic. We show that non-standard models of PA have Scott rank at least ω, but, other than that, there are no limits to their complexity. Given a completion T of PA we give a reduction via bi-interpretability of the class of linear orders to the models of T. This allows us to exhibit models of T of Scott rank α for every ω≤α≤ω1. In particular, every completion of T has models of high Scott rank.

This is joint work with Antonio Montalbán.

Speaker:     Gershom Bazerman, Arista Networks.

Date and Time:     Wednesday May 4, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Classes of Closed Monoidal Functors which Admit Infinite Traversals.

Stationary logic and set theory

Stationary logic was introduced in the 1970’s. It allows the quantifier 'for almost all countable subsets s…'. Although it is undoubtedly a kind of second order logic, it is completely axiomatizable, countably compact and satisfies a kind of Downward Lowenheim-Skolem theorem. In this talk I give first a general introduction to the extension of first order logic by this 'almost all'-quantifier. As 'almost all' is interpreted as 'for a club of', the theory of this logic is entangled with properties of stationary sets. I will give some examples of this. The main reason to focus on this logic in my talk is to use it to build an inner model of set theory. I will give a general introduction to this inner model, called C(aa), or the aa-model, and sketch a proof of CH in the model. My work on the aa-model is joint work with Juliette Kennedy and Menachem Magidor.

Michał Godziszewski, University of Vienna
Modal Quantifiers, Potential Infinity, and Yablo sequences

When properly arithmetized, Yablo's paradox results in a set of formulas which (with local disquotation in the background) turns out to be consistent, but ω-inconsistent. Adding either uniform disquotation or the ω-rule results in inconsistency. Since the paradox involves an infinite sequence of sentences, one might think that it doesn't arise in finitary contexts. We study whether it does. It turns out that the issue depends on how the finitistic approach is formalized. On one of them, proposed by Marcin Mostowski, all the paradoxical sentences simply fail to hold. This happens at a price: the underlying finitistic arithmetic itself is ω-inconsistent. Finally, when studied in the context of a finitistic approach which preserves the truth of standard arithmetic, the paradox strikes back - it does so with double force, for now the inconsistency can be obtained without the use of uniform disquotation or the ω-rule. This is joint work with Rafał Urbaniak from the University of Gdańsk.

The surprising strength of reflection in second-order set theory with abundant urelements

I shall give a general introduction to urelement set theory and the role of the second-order reflection principle in second-order urelement set theory GBCU and KMU. With the abundant atom axiom, asserting that the class of urelements greatly exceeds the class of pure sets, the second-order reflection principle implies the existence of a supercompact cardinal in an interpreted model of ZFC. The proof uses a reflection characterization of supercompactness: a cardinal κ is supercompact if and only if for every second-order sentence true in some structure M (of any size) is also true in a first-order elementary substructure m≺M of size less than κ. This is joint work with Bokai Yao. http://jdh.hamkins.org/surprising-strength-of-reflection-with-abundant-urelements-cuny-set-theory-seminar-april-2022

Speaker:     Morgan Rogers, Universit`a degli Studi dell’Insubria.

Date and Time:     Wednesday March 30, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Toposes of Topological Monoid Actions.

Abstract: Anyone encountering topos theory for the first time will be familiar with the fact that the category of actions of a monoid on sets is a special case of a presheaf topos. It turns out that if we equip the monoid with a topology and consider the subcategory of continuous actions, the result is still a Grothendieck topos. It is possible to characterize such toposes in terms of their points, and along the way extract canonical representing topological monoids, the complete monoids. I'll sketch the trajectory of this story, present some positive and negative results about Morita-equivalence of topological monoids, and explain how one can extract a semi-Galois theory from this set-up.

Speaker:     Joseph Dimos.

Date and Time:     Wednesday March 23, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Introduction to Fusion Categories and Some Applications.

Abstract: Tensor categories and multi-tensor categories have strong alignment with module categories. We can use the multi-tensor categories C in conjunction with classifying tensor algebras wrt C. From here, we can illustrate some examples of tensor categories: the category Vec of k-vector spaces that gives us a fusion category. This is defined as a category Rep(G) of some finite dimensional k-representations of a group G. From here, I will walk through the correspondence of tensor categories (Etingof) and fusion categories. Throughout, I will indicate a few unitary and non-unitary cases of fusion categories. Those unitary fusion categories are those that admit a uniquely monoidal structure. For example, this draws upon [Jones 1983] for finite index and finite depth that bridges a subfactor A-bimodule B to provide a full subcategory of some category A by its module structure. I will discuss some of these components throughout and explain the Morita equivalence of fusion categories.

Speaker:     Morgan Rogers, Universit`a degli Studi dell’Insubria.

Date and Time:     Wednesday March 30, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     TBA.

Models of Relevant Arithmetic

In the 1970s, the logician and philosopher Robert Meyer proposed a novel response to Goedel's Incompleteness Theorems, suggesting that perhaps the results' impact could be blunted by analyzing Peano arithmetic with a weaker deductive system. Initial successes of the program of relevant arithmetic were positive. E.g., R# (the theory of Peano arithmetic under the relevant logic R) can be shown consistent in the sense of not proving 0=1 and this can be shown through arguably finitistic methods. In this talk I will discuss the rise and fall of Meyer's program, detailing the philosophical foundations, its positive development, and the context of Harvey Friedman's negative result in 1992. I'll also suggest why the program, although not necessarily successful, is nevertheless an interesting object of study.

Also note that a great deal of context—including Meyer's two long-unpublished monographs on the topic—have recently appeared in a special issue of the Australasian Journal of Logic I co-edited with Graham Priest, which can be found at https://ojs.victoria.ac.nz/ajl/issue/view/751.

Speaker:     Joseph Dimos.

Date and Time:     Wednesday March 23, 2022, 7:00 - 8:30 PM., on Zoom.

Title:     Introduction to Fusion Categories and Some Applications.

Abstract: Tensor categories and multi-tensor categories have strong alignment with module categories. We can use the multi-tensor categories C in conjunction with classifying tensor algebras wrt C. From here, we can illustrate some examples of tensor categories: the category Vec of k-vector spaces that gives us a fusion category. This is defined as a category Rep(G) of some finite dimensional k-representations of a group G. From here, I will walk through the correspondence of tensor categories (Etingof) and fusion categories. Throughout, I will indicate a few unitary and non-unitary cases of fusion categories. Those unitary fusion categories are those that admit a uniquely monoidal structure. For example, this draws upon [Jones 1983] for finite index and finite depth that bridges a subfactor A-bimodule B to provide a full subcategory of some category A by its module structure. I will discuss some of these components throughout and explain the Morita equivalence of fusion categories.

Joel David Hamkins, Notre Dame University
Infinite wordle and the mastermind numbers

I shall introduce and consider the natural infinitary variations of Wordle, Absurdle, and Mastermind. Infinite Wordle extends the familiar finite game to infinite words and transfinite play—the code-breaker aims to discover a hidden codeword selected from a dictionary Δ⊆Σω of infinite words over a countable alphabet Σ by making a sequence of successive guesswords, receiving feedback after each guess concerning its accuracy. For any dictionary using the usual 26-letter alphabet, for example, the code-breaker can win in at most 26 guesses, and more generally in n guesses for alphabets of finite size n. Meanwhile, for some dictionaries on an infinite alphabet, infinite play is required, but the code-breaker can always win by stage ω on a countable alphabet, for any fixed dictionary. Infinite Mastermind, in contrast, is a subtler game than Wordle because only the number and not the position of correct bits is given. When duplication of colors is allowed, nevertheless, then the code-breaker can still always win by stage ω, but in the no-duplication variation, no countable number of guesses (even transfinite) is sufficient for the code-breaker to win. I therefore introduce the mastermind number, denoted mm, to be the size of the smallest winning no-duplication Mastermind guessing set, a new cardinal characteristic of the continuum, which I prove is bounded below by the additivity number add(M) of the meager ideal and bounded above by the covering number cov(M). In particular, the precise value of the mastermind number is independent of ZFC and can consistently be strictly between ℵ1 and the continuum 2ℵ0. In simplified Mastermind, where the feedback given at each stage includes only the numbers of correct and incorrect bits (omitting information about rearrangements), then the corresponding simplified mastermind number is exactly the eventually different number d(≠∗). http://jdh.hamkins.org/infinite-wordle-and-the-mastermind-numbers-cuny-logic-workshop-march-2022/

We investigate which forcing notions can be embedded into a Tree-Prikry forcing. It turns out that the answer changes drastically under different large cardinal assumptions. We will focus on the class of κ-strategically closed forcings of cardinality κ, <κ-strategically closed forcings of cardinality κ and the κ-distributive forcing notions of cardinality κ. Then we will examine distributive subforcings of the Prikry forcing of cardinality larger than κ. This is a joint work with Moti Gitik and Yair Hayut.

Joel David Hamkins, Notre Dame University
Infinite wordle and the mastermind numbers

I shall introduce and consider the natural infinitary variations of Wordle, Absurdle, and Mastermind. Infinite Wordle extends the familiar finite game to infinite words and transfinite play—the code-breaker aims to discover a hidden codeword selected from a dictionary Δ⊆Σω of infinite words over a countable alphabet Σ by making a sequence of successive guesswords, receiving feedback after each guess concerning its accuracy. For any dictionary using the usual 26-letter alphabet, for example, the code-breaker can win in at most 26 guesses, and more generally in n guesses for alphabets of finite size n. Meanwhile, for some dictionaries on an infinite alphabet, infinite play is required, but the code-breaker can always win by stage ω on a countable alphabet, for any fixed dictionary. Infinite Mastermind, in contrast, is a subtler game than Wordle because only the number and not the position of correct bits is given. When duplication of colors is allowed, nevertheless, then the code-breaker can still always win by stage ω, but in the no-duplication variation, no countable number of guesses (even transfinite) is sufficient for the code-breaker to win. I therefore introduce the mastermind number, denoted mm, to be the size of the smallest winning no-duplication Mastermind guessing set, a new cardinal characteristic of the continuum, which I prove is bounded below by the additivity number add(M) of the meager ideal and bounded above by the covering number cov(M). In particular, the precise value of the mastermind number is independent of ZFC and can consistently be strictly between ℵ1 and the continuum 2ℵ0. In simplified Mastermind, where the feedback given at each stage includes only the numbers of correct and incorrect bits (omitting information about rearrangements), then the corresponding simplified mastermind number is exactly the eventually different number d(≠∗). http://jdh.hamkins.org/infinite-wordle-and-the-mastermind-numbers-cuny-logic-workshop-march-2022/

Rasmus Blanck, University of Gothenburg
Incompleteness results for arithmetically definable extensions of strong fragments of PA

In this talk, I will present generalisations of some incompleteness results along two axes: r.e. theories are replaced by Σn+1-definable ones, and the base theory is pushed down as far as it will go below PA. Such results are often easy to prove from suitably formulated generalisations of facts used in the original proofs. I will present a handful of such facts, including versions of the arithmetised completeness theorem and the Orey–Hájek characterisation, to show what additional assumptions our theories must satisfy for the results to generalise. Two salient classes of theories emerge in this context: (a) Σn-sound extensions of IΣn + exp, and (b) Πn-complete, consistent extensions of IΣn+1. Finally, I will discuss some results that fail to generalise to Σn+1-definable theories, as well as an open problem related to Woodin's theorem on the universal algorithm.

The presentation is based on the following paper: https://doi.org/10.1017/S1755020321000307

Definable Well Orders and Other Beautiful Pathologies

Many sets of reals - well orders of the reals, MAD families, ultrafilters on omega etc - only necessarily exist under the axiom of choice. As such, it has been a perennial topic in descriptive set theory to try to understand when, if ever, such sets can be of low definitional complexity. Large cardinals rule out such the existence of projective well orders, MAD families etc while it's known that if V=L (or even just 'every real is constructible') then there is a Δ12 well order of the reals and Π11 witnesses to many other extremal sets of reals such as MAD families and ultrafilter bases. Recently a lot of work on the border of combinatorial and descriptive set theory has focused on considering what happens to the definitional complexity of such sets in models in which the reals have a richer structure - for instance when CH fails and various inequalities between cardinal characteristics is achieved. In this talk I will present a recent advance in this area by exhibiting a model where the continuum is ℵ2, there is a Δ13 well order of the reals, and a Π11 MAD family, a Π11 ultrafilter base for a P-point, and a Π11 maximal independent family, all of size ℵ1. These complexities are best possible for both the type of object and the cardinality hence this may be seen as a maximal model of 'minimal complexity witnesses'. This is joint work with Jeffrey Bergfalk and Vera Fischer.

Mauro di Nasso, Università di Pisa
Nonstandard natural numbers in arithmetic Ramsey Theory and topological dynamics

The use of nonstandard models *N of the natural numbers has recently found several applications in arithmetic Ramsey theory. The basic observation is that every infinite number in *N corresponds to an ultrafilter on N, and the algebra of ultrafilters is a really powerful tool in this field. Note that this notion also makes sense in any model of PA, where one can consider the 1-type of any infinite number.

Furthermore, nonstandard natural numbers are endowed with a natural compact topology, and one can apply the methods of topological dynamics considering the shift operator x↦x+1 . This very peculiar dynamic has interesting characteristics.

In this talk I will also present a new result in the style of Hindman’s Theorem about the existence of infinite monochromatic configurations in any finite coloring of the natural numbers. A typical example is the following monochromatic pattern:
a, b, c, … , a+b+ab, a+c+ac, b+c+bc, … , a+b+c+ab+ac+bc+abc.