# Models of PA

The Models of PA seminar meets regularly at the CUNY Graduate Center, holding talks on models of the Peano Axioms and related theories. It meets on (most) Mondays 6:30 - 8 PM at the CUNY Graduate Center in room 4214.03. It is organized by Roman Kossak and Erez Shochat.
(55 items)

Models of PAWednesday, April 1, 20154:50 pmGC 6300

# The undecidability of lattice-ordered groups II

GC CUNY

In this talk I will discuss conjugacy in the automorphism group of the rationals as a linearly ordered set, and then show that the integers, together with addition and multiplication, can be interpreted in Aut(Q), and thus that the theory of Aut(Q) is undecidable.

Wednesday, March 25, 20155:00 pmGC 7314

# Generic I0 at $\aleph_\omega$

Technische Universität Wien

It is common practice to consider the generic version of large cardinals defined with an elementary embedding, but what happens when such cardinals are really large? The talk will concern a form of generic I0 and the consequences of this over-the-top hypothesis on the “largeness” of the powerset of $\aleph_\omega$. This research is a result of discussions with Hugh Woodin.

Slides

Models of PAWednesday, March 18, 20154:50 pmGC 6300

# Countable Arithmetically Saturated Models and the Small Index Property.

St. Francis College

In 1994 Lascar proved that countable arithmetically saturated models of PA have the Small Index Property. In this talk we outline the proof and discuss related results and open problems.

Models of PAWednesday, March 11, 20154:50 pmGC 6300

# The undecidability of lattice-ordered groups

GC CUNY

In this talk I will discuss conjugacy in the automorphism group of the rationals as a linearly ordered set, and then show that the integers, together with addition and multiplication, can be interpreted in Aut(Q), and thus that the theory of Aut(Q) is undecidable.

Models of PAWednesday, March 4, 20154:50 pmGC 6300

# Definability in linear fragments of Peano arithmetic III

Charles University

In this series of talks, I will provide an analysis of definable sets in models of linear arithmetics. Here, for a cardinal k, the k-linear arithmetic (LA_k) is a full-induction arithmetical theory extending Presburger arithmetic by k non-standard scalars (= unary functions of multiplication by distinguished elements). Note that each model of a linear arithmetic naturally corresponds to a discretely ordered module over the ordered ring generated by the scalars.

I will prove a quantifier elimination result (QE up to disjunctions of bounded pp-formulas) for LA_1 and give a complete characterisation of definable sets in its models. On the other hand, I will construct an example of a model of LA_2 (or any LA_k with k at least 2) where multiplication is definable on a non-standard initial segment (and thus no pp-elimination is possible). Finally, I will show that all the theories LA_k satisfy quantifier elimination (at least) up to bounded formulas.

I will mention several applications and corollaries of these results including a construction of a non-NIP ordered module, or a result on automorphisms in saturated models of PA.

Models of PAWednesday, February 25, 20154:50 pmGC 6300

# Definability in linear fragments of Peano arithmetic II

Charles University

In this series of talks, I will provide an analysis of definable sets in models of linear arithmetics. Here, for a cardinal k, the k-linear arithmetic (LA_k) is a full-induction arithmetical theory extending Presburger arithmetic by k non-standard scalars (= unary functions of multiplication by distinguished elements). Note that each model of a linear arithmetic naturally corresponds to a discretely ordered module over the ordered ring generated by the scalars.

I will prove a quantifier elimination result (QE up to disjunctions of bounded pp-formulas) for LA_1 and give a complete characterisation of definable sets in its models. On the other hand, I will construct an example of a model of LA_2 (or any LA_k with k at least 2) where multiplication is definable on a non-standard initial segment (and thus no pp-elimination is possible). Finally, I will show that all the theories LA_k satisfy quantifier elimination (at least) up to bounded formulas.

I will mention several applications and corollaries of these results including a construction of a non-NIP ordered module, or a result on automorphisms in saturated models of PA.

Models of PAWednesday, February 18, 20154:50 pmGC 6300

# Definability in linear fragments of Peano arithmetic I

Charles University

In this series of talks, I will provide an analysis of definable sets in models of linear arithmetics. Here, for a cardinal k, the k-linear arithmetic (LA_k) is a full-induction arithmetical theory extending Presburger arithmetic by k non-standard scalars (= unary functions of multiplication by distinguished elements). Note that each model of a linear arithmetic naturally corresponds to a discretely ordered module over the ordered ring generated by the scalars.

I will prove a quantifier elimination result (QE up to disjunctions of bounded pp-formulas) for LA_1 and give a complete characterisation of definable sets in its models. On the other hand, I will construct an example of a model of LA_2 (or any LA_k with k at least 2) where multiplication is definable on a non-standard initial segment (and thus no pp-elimination is possible). Finally, I will show that all the theories LA_k satisfy quantifier elimination (at least) up to bounded formulas.

I will mention several applications and corollaries of these results including a construction of a non-NIP ordered module, or a result on automorphisms in saturated models of PA.

Models of PAWednesday, February 11, 20154:50 pmGC 6300

# Models with the omega-property

The City University of New York

A model M of PA has the omega-property if it has an elementary end extension coding a subset
of M of order type omega. The countable short recursively saturated models are a proper subclass
of the countable models with the omega-property, and both classes share many common
model theoretic properties. For example, they all have automorphism groups of size continuum. I will give a brief survey of what is known about models with the omega-property and I will discuss some open problems.

Models of PAWednesday, December 10, 20144:50 pmGC 6300

# Bounded Second-Order Arithmetic II

GC CUNY
Models of PAWednesday, December 3, 20144:50 pmGC 6300

# Bounded Second-Order Arithmetic I

GC CUNY

We will discuss axiom systems for weak second order arithmetic. In the study of bounded arithmetic, the base theory is called V^0; we will discuss coding and counting in V^0 and in an extension of it. We will give at least one example of a sentence that is not provable in V^0 but is provable in the extension.

Models of PAWednesday, November 19, 20144:50 pmGC 6300

# Forcing over models of arithmetic

I will talk about forcing over models of arithmetic. Our primary application will be the following theorem, due to Simpson: if a model M of PA is countable, then M has a subset U such that that (M,U) is a pointwise definable model of PA*. Time permitting, we will see that the MacDowell–Specker theorem fails for uncountable languages: for M countable and nonstandard, there are U_alpha for alpha < omega_1 such that (M, U_alpha)_{alpha < omega_1} is a model of PA* and has no elementary end extensions.

Models of PAWednesday, November 12, 20144:50 pmGC 6300

# Decoding in the automorphism group of a model of arithmetic.

Penn State Mont Alto

In the talk I will discuss results on recovering a countable recursively saturated model of Peano Arithmetic from its automorphism group.

Models of PAWednesday, November 5, 20144:50 pm

# The pentagon lattice III

Models of PAWednesday, October 29, 20144:50 pmGC 6300

# The pentagon lattice II

Models of PAWednesday, October 22, 20144:50 pmGC 6300

# The pentagon lattice

In this talk, I will discuss the proof that for every countable model M of PA, there is an end extension N such that Lt(N /M) is isomorphic to N5. I will begin by constructing an infinite representation of N5 and then discuss Theorem 4.5.21 in TSOMOPA which shows how to construct an extension from such a representation.

Models of PAWednesday, October 15, 20144:50 pmGC 6300

# The Hales-Jewett Theorem and Applications II

Bronx Community College
Models of PAWednesday, October 8, 20144:50 pmGC 6300

# The Hales-Jewett Theorem and Applications

Bronx Community College

The Hales-Jewett Theorem has applications in Ramsey Theory, and is also used to prove properties about lattices of elementary substructures. We will prove the Hales-Jewett Theorem and discuss applications.

Models of PAWednesday, October 1, 20144:50 pmGC 6300

# Elementary end extensions and the pentagon lattice

The City University of New York

By a theorem of Wilkie, every countable model M of PA has an elementary end extension N such that interstructure lattice Lt(N/N) is isomorphic to the pentagon lattice. I will explain why the theorem does not generalize to uncountable models.

Models of PAWednesday, September 17, 20144:50 pmGC 6300New location

# Ranked lattices and elementary end extensions

The City University of New York

Every countable model M of PA has and elementary end extension N such that the lattice Lt(N/M) is
isomorphic to the pentagon lattice N_5. I will go over the proof why none of such extensions can be conservative.

Models of PAMonday, May 12, 20147:00 pmGC 4214.03

# Lattices of elementary substructures of arithmetically saturated models of PA

GC CUNY
Models of PAMonday, May 12, 20146:00 pmGC 4214.03

# An introduction to arithmetically saturated models of PA

St. Francis College
Models of PAMonday, May 5, 20146:30 pmGC 4214.03

# Submodel Lattices of Nerode Semirings

University of Connecticut

Let TA be True Arithmetic, and let TA_2 be the set of Pi_2 sentences in TA.
If N is a model of TA_2, then the set Lt(N) of substructures of N that are also models of TA_2
forms a complete lattice. A Nerode semiring is a finitely generated model of TA_2.
I will be talking about some joint work with Volodya Shavrukov in which we investigate the possible lattices that are isomorphic to some Lt(N), where N is a Nerode semiring. Existentially closed models of TA_2 were studied long ago. The possible Lt(N) will also be considered for e.c. Nerode semirings.

Models of PAMonday, April 7, 20146:30 pm

# Lattices with congruence representations as interstructure lattices

The City University of New York

In this talk I will discuss arguably the most general result on which finite lattices may
arise as substructure lattices of models of Peano arithmetic. The focus will be on lattices with
congruence representations.

Let A be an algebra (a structure in a purely functional signature). We may consider the set of all congruence relations on A, which naturally forms a lattice Cg(A). A lattice L is said to have a congruence representation if there is an algebra A so that L is isomorphic to Cg(A)*. (Cg(A)* is the lattice obtained from Cg(A) be interchanging the roles of join and meet.) The main theorem is that if L is a finite lattice with a congruence representation and M is a non-standard model of PA then there is a cofinal elementary extension N of M so that the interstructure lattice Lt(N/M) is isomorphic to L.

This talk is intended as an overview. I do not plan on going into any details of the proofs but rather will survey the necessary tools that go into proving the theorem.

Models of PAMonday, March 31, 20146:30 pmGC 4214.03

# Finite Distributive Lattices II

I will continue the proof (in TSOMOPA 4.3) that if D is a finite distributive lattice, there is a model M such that Lt(M) is isomorphic to D.

Models of PAMonday, March 24, 20146:30 pmGC 4214.03

# Finite Distributive Lattices

In this talk, I will present the proof (in TSOMOPA 4.3) that if D is a finite distributive lattice, there is a model M such that Lt(M) is isomorphic to D.

Models of PAMonday, March 17, 20146:30 pmGC 4214.03

# Boolean algebras of elementary substructures II

The City University of New York

This is a continuation of the talk from last week. I will show how to use minimal types to construct elementary end extensions with large interstructure lattices.

Models of PAMonday, March 10, 20146:30 pmGC 4214.03

# Boolean algebras of elementary substructures

The City University of New York

In his 1976 paper Haim Gaifman proved that for every set I, every model M of PA has an elementary end extension N such that Lt(N/M) is isomorphic to P(I). I will present a proof.

Models of PAMonday, March 3, 20146:30 pmGC 4214.03

# Blass-Gaifman and Ehrenfeucht lemmas

GC CUNY

Proofs of the lemmas will be given and some consequences related to substructure lattices will be explained.

Models of PAMonday, February 24, 20146:30 pmGC 4214.03

# Introduction to Lattices and Substructure Lattices II

Bronx Community College

We will continue an introduction to Substructure Lattices, a theme for this semester’s seminar. It will still be completely elementary.

Models of PAMonday, February 10, 20146:30 pmGC 4214.03

# Introduction to Lattices and Substructure Lattices

Bronx Community College

This talk with be completely elementary. We will provide an introduction to Substructure Lattices, a theme for this semester’s seminar. Given a model M of Peano Arithmetic, its Substructure Lattice is the lattice of elementary substructures of M. We will discuss the basics of lattice theory relevant to understanding this topic and present some of the big questions in this area.

Models of PAWednesday, December 11, 20136:30 pmGC 4214.03

# Equivalence relations in models of Peano arithmetic

GC CUNY

The talk will be about the correspondence between definable equivalence relations on countable recursively saturated models of PA and the closed normal subgroups of their automorphism groups.

Models of PAWednesday, December 4, 20136:30 pmGC 4214.03

# When are subsets of a model “coded”? II

Models of PAWednesday, November 20, 20136:30 pmGC 4214.03

# When are subsets of a model “coded”?

I will present a result by J. Schmerl that characterizes when a collection of subsets of a given model, M, will appear as the coded sets in some elementary end extension of M. This is an analogue to Scott’s theorem, which characterizes when a collection of sets of natural numbers can be the standard system of some model of PA. If there is time, I will also present some extensions of the result.

Models of PAWednesday, November 13, 20136:30 pmGC 4214.03

# How to make a full satisfaction class

The City University of New York
Models of PAWednesday, November 6, 20136:30 pmGC 4214.03

# Tanaka’s embedding theorem

Bronx Community College
Models of PAWednesday, October 30, 20136:30 pmGC 4214.03

# Schmerl’s Lemma and Boundedly Saturated Models II

St. Francis College
Models of PAWednesday, October 23, 20136:30 pmGC 4214.03

# Schmerl’s Lemma and Boundedly Saturated Models

St. Francis College

We prove a slight modification of Schmerl’s Lemma for saturated models, and show how it can be applied to prove Kaye’s Theorem for boundedly saturated models of PA.

Models of PAWednesday, October 16, 20136:30 pmGC 4214.03

# Cofinal extensions of recursively saturated ordered structures

Queensborough Community College, CUNY
Models of PAWednesday, October 9, 20136:30 pmGC 4214.03

# More on fullness

Continuing with the discussion from last week, I will state a few conditions that imply fullness and use that to show a few basic examples of full models. I will also show one direction of Kaye’s theorem that a model M is full if and only if its standard system is a model of full second order comprehension (CA_0).

Models of PAWednesday, October 2, 20136:30 pmGC 4214.03

# Fullness

The City University of New York

A model $M$ of PA is full if for every definable in $(M,omega)$ set $X$, $Xcap omega$ is coded in $M$. In a recent paper, Richard Kaye proved that $M$ is full if and only if its standard system is a model of full second order comprehension. Later in the semester we will examine Kaye’s proof. In this talk I will discuss some preliminary results and I will show an example of a model that is not full, using an argument that does not depend on Kaye’s theorem

Models of PAWednesday, May 8, 20135:00 pmGC 4214.03

# The automorphism group of a model of arithmetic: recognizing standard system

Penn State Mont Alto

Let M be countable recursively saturated model of Peano Arithmetic. In the talk I will discuss ongoing research on recognizing standard system of M in the automorphism group of M.

Models of PAWednesday, April 24, 20136:30 pmGC 4214.03

# Regular Interstices

St. Francis College

We define the notion of a regular interstice and show that every regular interstice has elements realizing selective types.

Models of PAWednesday, April 17, 20136:30 pmGC 4214.03

# Pseudostandard cuts

The City University of New York

A cut I in a model M of PA is pseudostandrd if there is an N such that (M,I) is elementary
equivalent to (N,omega). I will discuss some preliminary results in model theory of pseudostandard cuts.

Models of PAWednesday, March 20, 20136:45 pmGC 4214.03

# Fast Growing Functions and Arithmetical Independence Results

The Leeds Logic Group, University of Leeds

We explore the role of the function $a+2^x$ and its generalisations to higher number classes, in supplying complexity bounds for the provably computable functions across a broad spectrum of (arithmetically based) theories. We show how the resulting “fast growing” subrecursive hierarchy forges direct links between proof theory and various combinatorial independence results – e.g. Goodstein’s Theorem (for Peano Arithmetic) and Friedman’s Miniaturised Kruskal Theorem for Labelled Trees (for $Pi^1_1$-CA$_0$).

Ref: Schwichtenberg and Wainer, “Proofs and Computations”, Persp. in Logic, CUP 2012.

Models of PAWednesday, March 13, 20138:00 amGC 4214.03

# Generalizing the notion of interstices

Ghent University

I will present a generalization of the notion of interstices that
originated from the study of generic cuts.

Models of PAWednesday, March 6, 20136:30 pmGC 4214.03

# Several versions of self-embedding theorem

Mathematical Institute, Tohoku University

In this talk, I will give several versions of Friedman’s self-embedding theorem which can characterize subsystems of Peano arithmetic. Similarly, I will also give several variations of Tanaka’s self-embedding theorem to characterize subsystems of second-order arithmetic.

Models of PAWednesday, February 27, 20136:30 pmGC 4214.03

# Introduction to interstices and intersticial gaps II

St. Francis College
Models of PAWednesday, February 20, 20136:30 pmCUNY Graduate Center in room 4214.03.

# Introduction to interstices and intersticial gaps

St. Francis College

Let M be a model of PA for which Th(M) is not Th(N) (N is the standard model). Then M has nonstandard definable elements. Let c be a non-definable element. The largest convex set which contains c and no definable elements is called the interstice around c. In this talk we discuss various properties of interstices. We also define intersticial gaps which are special subsets of interstices. We show that the set of the intersticial gaps which are contained in any given interstice of a countable arithmetically saturated model of PA is a dense linear order.