NYLogic

Given a model M of an axiomatic theory A, and a model N of an axiomatic theory B, we say that they are bi-interpretable if, roughly speaking, they have the same definable sets: that is, there are definable maps that move definable sets in one to definable sets in the other. One interesting question we might ask, given an axiomatic theory A, is which of its models are bi-interpretable with the integers (seen as a model of the first-order theory of rings)? As self-interpretations of the integers are particularly simple, this gives a lot of information about properties of the model. In this talk, we will consider arithmetic groups like SL(n, Z) and discuss recent progress in understanding when such groups are bi-interpretable with arithmetic and what consequences this has when it occurs.

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Nonstandard methods have been successfully applied to standard problems in number theory by R. Jin, T. Tao and others. A. Boudaoud and D. Bellaouar are pursuing the opposite direction: they are formulating number-theoretic problems in the language of nonstandard analysis and solving them by standard methods. Two examples of the kind of questions they consider are:
(1) Can every unlimited natural number n be represented in the form n = s + w_1w_2 where s is a limited integer and w_1, w_2 are unlimited?
(2) Can every unlimited natural number n be represented in the form n = w_1w_2 + w_3w_4 so that each ratio w_i / w_j is appreciable (ie, neither infinitesimal nor unlimited)?
I give a negative answer to question (1) (assuming Dickson’s Conjecture) and a positive answer to question (2).
A. Boudaoud, D. Bellaouar, Representation of integers: A nonclassical point of view, Journal of Logic & Analysis. 12:4 (2020) 1{31; K. Hrbacek, Journal of Logic & Analysis 12:5 (2020) 1–6.

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It is a well-known fact that non-standard models of Peano Arithmetic (PA) have order type N + Z · D, where D is a dense linear order. The question of which dense linear orders D can occur in such order types is non-trivial and widely studied. In this context Friedman asked the following question:

Are there consistent extensions of Peano Arithmetic T and T′ such that the class of order types of models of T and the class of order types of models of T′ differ?

Friedman’s question is very complex and still wide open. In this talk we will go in the opposite direction and consider a version of Friedman’s question for syntactic fragments of PA. We will present results from a joint work with Benedikt Löwe on order types of non-standard models of syntactic subsystems of arithmetic obtained by restricting the language to subsets of the operations. We will put particular emphasis on models of syntactic subsystems of Peano Arithmetic obtained by dropping the schema of induction.

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I call a subset of the domain of a countable model absolutely undefinable if the set of its images under automorphisms of the model is uncountable. By the Kueker-Reyes theorem, all sets that are not absolutely undefinable are parametrically definable in $L_{\omega_1 \omega}$. I will survey classical results about first-order undefinability in the standard model of arithmetic, and I will contrast them with some old and some new results about absolute undefinability in nonstandard models of PA.​

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A typical Ramsey statement is the following: given a coloring of a complete graph, we aim to find a 'large' complete subgraph that is monochromatic. The weaker variation we are considering here (introduced by Bergfalk-Hrusak-Shelah) is to relax the 'complete subgraph' in the goal. More precisely, we aim to find a certain 'large' connected monochromatic subgraph. We will discuss the motivation and the connections with other combinatorial and algebraic problems. We demonstrate consistently, such partition relations can hold at small uncountable cardinals like aleph_2, and successors of singular cardinals like aleph_{omega+1}. Joint work with Hrusak and Shelah.

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We survey the effective foundations for analysis with infinitesimals recently developed by Hrbacek and Katz, and detail some applications. Theories SPOT and SCOT illustrate the fact that analysis with infinitesimals requires no more choice than traditional analysis. The theory SCOT incorporates in particular all the axioms of Nelson's Radically Elementary Probability Theory, which is therefore conservative over ZF+ADC.

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It is well known that a countable model of PA has a truth predicate if and only if it is recursively saturated. It is also well known that not all countable recursively saturated models of PA have *inductive* or even $\Delta_0$-inductive truth predicates: indeed, such models must satisfy Con(PA), for example. Recent work by Enayat-Pakhomov and Cieśliński-Łełyk-Wcisło explored the principle of 'disjunctive correctness', asserting that every disjunction is true if and only if it has a true disjunct. In particular, one can show that every countable model of PA has a 'disjunctively trivial' elementary extension: that is, an elementary extension with a truth predicate in which all nonstandard length disjunctions are evaluated as true. In this talk, we will see that such 'disjunctively trivial' models are necessarily arithmetically saturated; indeed, we will see that a countable model of PA is arithmetically saturated if and only if it has a disjunctively trivial truth predicate. We will explore related pathologies in truth predicates, and classify the sets which can be defined using such pathologies. We find other surprising connections between arithmetic saturation and these questions of definability. This is joint work with Mateusz Łełyk, based heavily on unpublished work by Jim Schmerl.

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The popular game of Hex can be extended to the infinite hexagonal lattice, defining a winning condition which formalises the idea of a chain of colored stones stretching towards infinity. The descriptive-set-theoretic complexity of the set of winning positions is unknown, although it is at most Σ^1_1, and it is conjectured to be Borel; this has implications on whether games of infinite Hex are determined from all initial positions as either first-player wins or draws.

I will show that, unlike the finite game, infinite Hex with an initially empty board is a draw. But is the game still a draw when starting from a non-empty board? This open question can be partially answered in the positive by assuming the existence of certain local strategies, and in the negative by giving the advantage of placing two stones at each turn to one of the players. This is joint work with Joel David Hamkins.

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Parameters are free variables in various axiom schemata in PA, ZFC, and other similar theories. Given an axiom schema S, we let S* be the parameter-free sub-schema.

Kreisel (A survey of proof theory, JSL 1968) was one of those who paid attention to the comparison of some schemata in second-order PA and their parameter-free versions. In particular, Kreisel noted that

[...] if one is convinced of the significance of something like a given axiom schema, it is natural to study details, such as the effect of parameters.

This talk is devoted to the effect of parameters in the schemata of Comprehension and Choice in second-order arithmetic.

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Fix a computable presentation $\overline{\mathbb Q}$ of the algebraic closure of the rational numbers. The absolute Galois group of the rational numbers, which is precisely the automorphism group of the field $\overline{\mathbb Q}$, may then be viewed as a collection of paths through a finite-branching tree. Each individual automorphism has a Turing degree. We will use known results in computability to try to build natural countable elementary subgroups of the absolute Galois group. Several intriguing questions in number theory will appear as we measure the extent to which we succeed in doing so.

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In this talk I will discuss some recent advances in the first-order classification problem. I will touch on first-order rigidity and quasi finite axiomatization. However, the main point of the presentation is on how, in principle, one can describe all structures which are first-order equivalent to a given one. This leads to non-standard models of algebraic structures (aka non-standard analysis or non-standard arithmetic), which are interesting in their own right.

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