September 2
Petr Glivický,
Universität Salzburg
The $\omega$-iterated nonstandard extension of $\mathbb{N}$ and Ramsey combinatorics
In the theory of nonstandard methods (traditionally known as nonstandard analysis), each mathematical object (a set) $x$ has a uniquely determined so called nonstandard extension ${}^*x$. In general, ${}^*x \supsetneq \{{}^*y; y\in x\}$ - that is, besides the original 'standard' elements ${}^*y$ for $y\in x$, the set ${}^*x$ contains some new 'nonstandard' elements.
For instance, some of the nonstandard elements of ${}^*\mathbb{R}$ can be interpreted as infinitesimals (there is $\varepsilon\in {}^*\mathbb{R}$ such that $0<\varepsilon<1/n$ for all $n\in\mathbb{N}$) allowing for nonstandard analysis to be developed in ${}^*\mathbb{R}$, while ${}^*\mathbb{N}$ turns out to be an (at least $\aleph_1$-saturated) nonstandard elementary extension of $\mathbb{N}$ (in the language of arithmetic).
While the whole nonstandard real analysis is most naturally developed in ${}^*\mathbb{R}$ (with just a few advanced topics where using the second extension ${}^{**}\mathbb{R}$ is convenient, though far from necessary), recent successful applications of nonstandard methods in combinatorics on $\mathbb{N}$ have utilized also higher order extensions ${}^{(n)*}\mathbb{N} = {}^{***\cdots *}\mathbb{N}$ with the chain $***\cdots *$ of length $n>2$.
In this talk we are going to study the structure of the $\omega$-iterated nonstandard extension ${}^{\cdot}\mathbb{N} = \bigcup_{n\in\omega} {}^{(n)*}\mathbb{N}$ of $\mathbb{N}$ and show how the obtained results shed new light on the complexities of Ramsey combinatorics on $\mathbb{N}$ and allow us to drastically simplify proofs of many advanced Ramsey type theorems such as Hindmann's or Milliken's and Taylor's.