David Ballard: *Lessons from Foundational Work in Nonstandard Set
Theory: *

a Set Theoretic Cosmology for Mathematics.

#### This talk will offer a preliminary report on a recent initiative, begun in
collaboration with Karel Hrbacek, which seeks to unite the lessons of
foundational work in both standard and nonstandard set theory. The modest goal
is a grand theory of everything.

### Mauro Di Nasso: *A Set Theory Where all Sets Have Nonstandard
Extensions.*

#### We present a set theory ZFC[Omega] where all sets have nonstandard
extensions. It is finitely axiomatized over ZFC-, and provides a k+ saturated
elementary embedding of the universe for each infinite cardinal k. In the
countable case, ZFC[Omega] includes the "alpha-theory" (joint work with V.
Benci), which provides a truly elementary approach to nonstandard analysis.
Every model of ZFC is the wellfounded part of some model of ZFC[Omega]. In
particular, the two theories are equiconsistent.

### Ali Enayat: *Conservative Extensions of Models of Arithmetic and Set
Theory.*

#### A model N is said to be a conservative extension of a model M if the
intersection of every parametrically definable subset of N with M is also
parametrically definable in M. This notion plays a key role in
the

model theory of arithmetic, and was initially introduced by Gaifman
and Phillips in relation to the McDowell-Specker theorem. In this talk we
compare/contrast the behavior of models of arithmetic and set theory in the
context of conservative extensions.

### Harvey Friedman: *Two Universe Axiomatizations of Set
Theory.*

### E.I. Gordon (joint work with P.V. Andreyev): *An Axiomatics for
Nonstandard Set Theory.*

#### We present an axiomatic framework for nonstandard analysis - the Nonstandard
Class Theory NCT which extends von Neumann - Goedel - Bernays Set Theory NBG by
adding a unary predicate symbol St to the language of NBG (St(X) means that the
class X is standard) and axioms - related to it - analogs of Nelson's
idealization, standardization and transfer principles. Those principles are
formulated as axioms, rather than axiom schemes, so that NCT is finitely
axiomatizable. NCT can be considered as a theory of definable classes of Bounded
Set Theory by V. Kanovei and M. Reeken. In many aspects NCT resembles the
Alternative Set Theory by P. Vopenka. For example there exist semisets (proper
subclasses of sets) in NCT and it can be proved that a set has a standard finite
cardinality iff it does not contain any proper subsemiset. Semisets can be
considered as external classes in NCT. Thus the saturation principle can be
formalized in NCT. We introduced also the Theory of Hyperfinite Sets (THS), that
is obtained from NCT by replacement of the axiom of infinity by its negation and
discuss some problems, concerning the approximation of infinite structures by
finite ones that can be formalized in NCT and THS.

####

### Karel Hrbacek: *Nonstandard Set Theory and Some of its Problems.*

#### Nonstandard set theory aims to axiomatize concepts and practices of concern
to nonstandard analysts. The talk will survey the difficulties encountered by,
attempts to adequately capture "nonstandard intuition", and some of the proposed
solutions. Related technical problems, and their relevance to standard set
theory, will also be discussed.

####

### Renling Jin: Picking One Point in Every Monad Countably Determinedly.

#### Let *Z be the set of all integers in a nonstandard universe and let U be a
cut (a cut is an infinite proper initial segment of all non-negative integers in
*Z and closed under addition). A cut U is an external set. For any number a in
*Z, the U-monad of a is the set of all integers in *Z such that the distance
between any of these integers and a is in U. The topic I like to present is
whether one can have a countably determined set which contains exactly one
element in each U-monad for some cut U (I like to call this kind of set a
U-choice set). Keisler and Leth showed that there is no analytic U-choice set
for any cut U and there is no countably determined U-choice set if U has an
uncountable cofinality. Several years ago, Kanovei asked me if there could be a
countably determined U-choice set for U being the set of all standard natural
numbers. In the talk, I will present a complete answer to the question for every
cut U with countable cofinality.

### Roman Kossak: *Nonstandard Model Theory?*

#### In his 1962 paper "On languages which are based on non-standard arithmetic"
Abraham Robinson initiated the study of semantics of languages obtained by
replacing the natural numbers as a base in the formulation of logical syntax by
a non-standard model of arithmetic. About twenty years later Robinson's ideas
were developed by Kotlarski, Krajewski, Lachlan, and Smith into the theory of
nonstandard satisfaction classes. I will give a survey of the main results,
concentrating on the question: How much of the model theory of arithmetic can be
reduced to the study of nonstandard satisfaction classes?

####

### David Ross: *Ramsey Theory and Nonstandard Analysis.*

#### I will survey some applications of Ramsey-type results within nonstandard
analysis, as well as some nonstandard approaches to Ramsey theory.

### James Schmerl: *Automorphisms of Models of Peano Arithmetic.*

#### I will discuss automorphisms of models M of Peano Arithmetic and also
Aut(M), the group of automorphisms of M. The main results to be considered is
that a group G is isomorphic to some Aut(M) iff G is
right-orderable.