Models of PA seminar

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.
(36 items)

Models of PAMonday, May 12, 20146:00 pmGC 4214.03
Erez Shochat

An introduction to arithmetically saturated models of PA

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

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
Alf Dolich

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
Athar Abdul-Quader

Finite Distributive Lattices II

The CUNY Graduate Center

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
Athar Abdul-Quader

Finite Distributive Lattices

The CUNY Graduate Center

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
Roman Kossak

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
Roman Kossak

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
Kerry Ojakian

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
Kerry Ojakian

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
Simon Heller

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
Athar Abdul-Quader

When are subsets of a model “coded”? II

The CUNY Graduate Center
Models of PAWednesday, November 20, 20136:30 pmGC 4214.03
Athar Abdul-Quader

When are subsets of a model “coded”?

The CUNY Graduate Center

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
Alf Dolich

How to make a full satisfaction class

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

Tanaka’s embedding theorem

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

Schmerl’s Lemma and Boundedly Saturated Models II

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

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
Whanki Lee

Cofinal extensions of recursively saturated ordered structures

Queensborough Community College, CUNY
Models of PAWednesday, October 9, 20136:30 pmGC 4214.03
Athar Abdul-Quader

More on fullness

The CUNY Graduate Center

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
Roman Kossak

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
Ermek Nurkhaidarov

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
Erez Shochat

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
Roman Kossak

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
Stan Wainer

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
Tin Lok Wong

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
Keita Yokoyama

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 20, 20136:30 pmCUNY Graduate Center in room 4214.03.
Erez Shochat

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.