In 1973 Abraham Robinson gave a talk about the nonstandard analysis (NSA) at the Institute for Advanced Study. After his talk Kurt Gödel made a comment, in which he predicted that “…there are good reasons to believe that Non-Standard Analysis in some version or other will be the analysis of the future”. One has to admit that during almost forty five years since this prediction was made, it did not come true. Although the NSA simplified proofs of many deep results in standard mathematics and allowed to obtain new standard results, among which there are some long standing open problems, it did not become the working tool for the most part of mathematicians. When they are interested in some result obtained with the help of the NSA, they prefer to reprove it in standard terms. One of the reasons of rejecting the NSA, is that as a rule the job of reproving is not difficult. The other reason is that the transfer principle of the NSA that is crucial for deduction of standard results from nonstandard ones relies significantly on formalization of mathematics in the the framework of superstructures or of the Axiomatic Set Theory.
For mathematicians working in ODE, PDE and other areas oriented toward applications, who use at most the naïve set theory, these formal languages may be difficult and irrelevant, so they may not feel confident in nonstandard proofs.

In this talk I will present a new version of nonstandard set theory, that is formulated on the same level of formalization as the naïve set theory. I will try to justify my opinion that this is a version, in which the nonstandard analysis may become the analysis of the future. I will discuss some examples of NSA theorems about interaction between some statements in continuous and their computer simulations that are rigorous theorems in the NSA but can not be formulated in terms of standard mathematics. These theorems have clear intuitive sense and even can be monitored in computer experiments. Nowadays many applied mathematicians share a point of view that the continuous mathematics is an approximation of the discrete one but not vice versa. This point of view can be easy formalized in the naïve nonstandard set theory above. Being not interested in proving classical theorems with the help of NSA, we don’t need the predicate of standardness and the Transfer Principle of the NSA in full. This allows to avoid an excessive formalization.

Nonstandard Analysis (NSA) was introduced around 1965 by Robinson as a formalization of the intuitive infinitesimal calculus which is in use to date in most of physics and historically in mathematics until the advent of Weierstrass’ epsilon-delta framework. Famous people like Connes and Bishop have derided NSA for its alleged utter lack of computational/effective/constructive content. In this talk I show that every theorem of ‘pure’ NSA can be (equivalently) converted to a theorem of computable mathematics. In many cases, the resulting theorem is even constructive in the sense of Bishop.

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.

Classically, the existence of an object tells us very little about how to construct said object.
We consider a nonstandard version of Ulrich Kohlenbach’s higher-order Reverse Mathematics
in which there is a very elegant and direct correspondence between, on one hand, the existence
of a functional computing an object and, on the other hand, the classical existence of this object
with the same standard and nonstandard properties. We discuss how these results -potentially-
contribute to the programs of finitistic and predicativist mathematics.

Nonstandard set theory enriches the usual set theory by a unary “standardness” predicate.  Investigations of its foundations raise a number of questions that can be formulated in ZFC or GB and appear open.  I will discuss several such problems concerning elementary embeddings, ultraproducts, ultrafilters and large cardinals.

This talk will be part of CANT 2013, the Combinatorial and Additive Number Theory Conference, on May 21-24, 2013 at the Graduate Center.

Abstract: By using nonstandard analysis, and in particular iterated elementary (nonstandard) extensions, we give foundations to a peculiar way of manipulating idempotent ultrafilters. The resulting formalism is suitable for applications in Ram- sey theory of numbers. To illustrate the use of this technique, we give (rather) short proofs of two important results in combinatorial number theory, namely Milliken- Taylor’s Theorem (a generalization of Hindman’s theorem), and Rado’s theorem about partition regularity of diophantine equations, in a new version formulated in terms of idempotent ultrafilters.
Some familiarity with the notion of elementary extension will be assumed in the first part of the talk, but in the second part about applications I will not assume any specific prerequisite (also the notions of ultrafilter and of partition regularity will be recalled).