# Models of PA

# 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.

# The Hales-Jewett Theorem and Applications II

# The Hales-Jewett Theorem and Applications

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.

# Elementary end extensions and the pentagon lattice

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.

# Ranked lattices and elementary end extensions

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.

# Lattices of elementary substructures of arithmetically saturated models of PA

# An introduction to arithmetically saturated models of PA

# Submodel Lattices of Nerode Semirings

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.

# Lattices with congruence representations as interstructure lattices

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.

# 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.

# 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.

# Boolean algebras of elementary substructures II

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.

# Boolean algebras of elementary substructures

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.

# Blass-Gaifman and Ehrenfeucht lemmas

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

# Introduction to Lattices and Substructure Lattices II

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

# Introduction to Lattices and Substructure Lattices

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.

# Equivalence relations in models of Peano arithmetic

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.

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

# 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.

# How to make a full satisfaction class

# Tanaka’s embedding theorem

# Schmerl’s Lemma and Boundedly Saturated Models II

# Schmerl’s Lemma and Boundedly Saturated Models

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.

# Cofinal extensions of recursively saturated ordered structures

# 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).

# Fullness

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

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

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.

# Regular Interstices

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

# Pseudostandard cuts

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.

# Fast Growing Functions and Arithmetical Independence Results

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.

# Generalizing the notion of interstices

I will present a generalization of the notion of interstices that

originated from the study of generic cuts.

# Several versions of self-embedding theorem

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.

# Introduction to interstices and intersticial gaps II

# Introduction to interstices and intersticial gaps

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.