# Blog Archives

These are the items posted in this seminar, currently ordered by their post-date, rather than by the event date. We will create improved views in the future. In the meantime, please click on the Seminar menu item above to find the page associated with this seminar, which does have a more useful view order.# Automorphisms of models of Presburger arithmetic II

# Automorphisms of models of Presburger arithmetic I

I will present an algebraic construction of end extensions of models of Presburger arithmetic, as well as key definitions and examples that are needed for studying the automorphism group of certain countable models of Pr.

# Loftiness V

A class C of countable models of PA is countably PC*_δ if there is a theory T in a countable language extending PA* such that for every countable model M of PA, M is in C if and only M is expandable to a model of T. The class of countable recursively saturated models of PA is countably PC*_δ, and Kaufman and Schmerl showed that many other natural classes, including the class uniformly ω-lofty models, are not. I will go over the proof of the Kaufman-Schmerl result, and I will discuss other potential approaches to characterizing classes of models of PA via expandability.

# Substructure Lattices of Recursively Saturated Models

Previously, it was known (Kossak-Schmerl 1995) that given two arithmetically saturated models with isomorphic substructure lattices, their standard systems must be the same. In Schmerl’s new paper (2015?), he analyzes this proof a little more carefully: in particular, if M is recursively saturated, and X subset omega, then there is an ideal in the substructure lattice of M corresponding to X if and only if X is Turing computable from the jump of some Y in SSy(M). We will go over this proof and, time permitting, how this is used in the proof of the main theorem: if M and N are countable arithmetically saturated models with isomorphic automorphism groups, then the jumps of their theories are Turing equivalent.

# Cofinal elementary extensions II

We will also discuss K-tallness and tallness in relation to the existence of *κ*-pseudosaturated cofinal elementary extensions. Here, *κ*-pseudosaturated structures are the structures all of whose infinite definable sets have a cardinality ≥ *κ*.

# Cofinal elementary extensions

We will discuss two weak versions of recursive saturation: K-tallness and tallness. K-tallness characterizes the countable models of *IΔ _{0} + exp* that have cofinal elementary extensions enlarging any infinite definable set, and tallness characterizes such models of linear ordering without the last element. I will present the proof of the latter. If time permits, we will also discuss the two properties in relation to the existence of

*κ*-pseudosaturated cofinal elementary extensions. Here,

*κ*-pseudosaturated structures are the structures all of whose infinite definable sets have a cardinality

*≥κ*.

# Recursive definability of the standard cut

Say that the standard cut in a model of arithmetic is recursively definable if there is a recursive sequence coinitial in the nonstandard elements. We will construct minimal models in which the standard cut is recursively definable and minimal models in which the standard cut is not recursively definable.

# Loftiness IV

I will discuss more work of Kaufmann and Schmerl around loftiness. In particular I will discuss how in the definition of e-loftiness we may restrict our attention to only those types that define cuts. These consideration lead to a simple proof of a theorem of Pabion’s that for kappa an uncountable cardinal a model M of PA is kappa-saturated if and only if its underlying ordering is kappa-saturated. Time permitting I will also discuss how for countable models M, being lofty is equivalent to having a recursively saturated simple extension.

# Loftiness III

A model M of PA has the $omega$-property if it has an elementary end extension coding a subset of M of order type $omega$. Tall models with the $omega$-property are uniformly $omega$-lofty. I will present several results on models with the $omega$-property, including Jim Schmerl’s construction of a model with the $omega$-property that is not recursively saturated.

# Loftiness II

I will introduce uniformly omega-lofty models and I will discuss some results about them due to Matt Kaufmann and Jim Schmerl.

# Loftiness

Loftiness is a weak version of recursive saturation. It was introduced and studied by Kaufmann and Schmerl in two papers published in 1984 and 1987. I will present basic definitions and results leading to constructions of lofty models of PA that are not recursively saturated.

# Enayat models II

Simpson proved that every countable model of PA has an expansion (to PA*) that is pointwise definable. The natural question, then, is if every countable model has an expansion to PA* in which no new elements are defined. Enayat proved this is false by showing the existence of many models that are not pointwise definable, but become so upon addition of any undefinable class. Inspired by this, I have begun thinking about which models have this property. I will describe some models with this property (and some without) and talk about my search for a non-trivial such model (I will also explain what I mean by “non-trivial” here).

# Enayat Models

Simpson proved that every countable model of PA has an expansion (to PA*) that is pointwise definable. The natural question, then, is if every countable model has an expansion to PA* in which no new elements are defined. Enayat proved this is false by showing the existence of many models that are not pointwise definable, but become so upon addition of any undefinable class. Inspired by this, I have begun thinking about which models have this property. I will describe some models with this property (and some without) and talk about my search for a non-trivial such model (I will also explain what I mean by “non-trivial” here).

# Ultrafilters and nonstandard methods in combinatorics of numbers

In certain areas of Ramsey theory and combinatorics of numbers, diverse non-elementary methods are successfully applied, including ergodic theory, Fourier analysis, (discrete) topological dynamics, algebra in the space of ultrafilters. In this talk I will survey some recent results that have been obtained by using tools from mathematical logic, namely ultrafilters and nonstandard models of the integers.

On the side of Ramsey theory, I will show how the hypernatural numbers of nonstandard analysis can play the role of ultrafilters, and provide a convenient setting for the study of partition regularity problems of diophantine equations. About additive number theory, I will show how the methods of nonstandard analysis can be used to prove density-dependent properties of sets of integers. A recent example is the following theorem: If a set $A$ of natural numbers has positive upper asymptotic density then there exists infinite sets $B$, $C$ such that their sumset $C+B$ is contained in the union of $A$ and a shift of $A$. (This gives a partial answer to an old question by Erdős.)

The slides are here.

# Linear fragments of PA – beyond LA_1

I will continue my series of talks on linear fragments of PA. Nevertheless, this talk will be mostly self contained.

In my previous talks, I paid attention almost exclusively to the case of the linear arithmetic LA_1 – a reduct of PA containing only unary multiplication by one distinguished element (scalar) instead of the complete binary multiplication. It was shown that LA_1 is a rather tame theory with most of the properties similar to those of Presburger arithmetic (model completeness, decidability, existence of non-standard recursive models, NIP, …).

In this talk, I will start to examine linear arithmetics with more then one scalar. Starting with LA_2 (two scalars) already, the picture changes dramatically – I will construct a model of LA_2 where a “Peano multiplication” is definable on an infinite initial segment. On the other hand, the quantifier elimination does not fail completely in linear arithmetics with more than one scalar – I will show that all linear arithmetics satisfy bounded QE.

I will summarize the results of this and previous talks in an almost complete characterization of linear fragments of PA which are/aren’t model complete.

If time permits, I will show some applications (in particular a result on (in)dependence of values of multiplication in different points in a saturated model of PA).

# Order Type vs. Isomorphism Type for Models of PA II

I will continue to discuss work of Shelah on the problem of under what conditions can the underlying order

type of a model of PA determine its isomorphism type. In particular I will outline a proof of a theorem roughly

stating that a model, M, of PA whose underlying order exhibits a week form of rigidity has the property that any

other model of PA which is order isomorphic to M is in fact isomorphic (as models of PA) to M.

# A perfectly generic talk

We will look at perfect generics for countable models of PA. Our goal will be to prove the following theorems: 1. Any countable collection of inductive subsets of a countable model are definable from a single generic. 2. Any countable model has minimally undefinable generics.

# Order type vs. Isomorphism type for models of PA

I will discuss recent work of Shelah around the general problem of how the underlying order type of a model of PA may determine influence its isomorphism type as a full model of PA.

# The undecidability of lattice-ordered groups II

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.

# Countable Arithmetically Saturated Models and the Small Index Property.

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.