1,870 research outputs found
Finite Embeddability of Sets and Ultrafilters
A set A of natural numbers is finitely embeddable in another such set B if
every finite subset of A has a rightward translate that is a subset of B. This
notion of finite embeddability arose in combinatorial number theory, but in
this paper we study it in its own right. We also study a related notion of
finite embeddability of ultrafilters on the natural numbers. Among other
results, we obtain connections between finite embeddability and the algebraic
and topological structure of the Stone-Cech compactification of the discrete
space of natural numbers. We also obtain connections with nonstandard models of
arithmetic.Comment: to appear in Bulletin of the Polish Academy of Sciences, Math Serie
On algorithm and robustness in a non-standard sense
In this paper, we investigate the invariance properties, i.e. robust- ness, of phenomena related to the notions of algorithm, finite procedure and explicit construction. First of all, we provide two examples of objects for which small changes completely change their (non)computational behavior. We then isolate robust phenomena in two disciplines related to computability
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom
Ultrafilters maximal for finite embeddability
In [1] the authors showed some basic properties of a pre-order that arose in
combinatorial number theory, namely the finite embeddability between sets of
natural numbers, and they presented its generalization to ultrafilters, which
is related to the algebraical and topological structure of the Stone-\v{C}ech
compactification of the discrete space of natural numbers. In this present
paper we continue the study of these pre-orders. In particular, we prove that
there exist ultrafilters maximal for finite embeddability, and we show that the
set of such ultrafilters is the closure of the minimal bilateral ideal in the
semigroup (\bN,\oplus), namely \overline{K(\bN,\oplus)}. As a consequence,
we easily derive many combinatorial properties of ultrafilters in
\overline{K(\bN,\oplus)}. We also give an alternative proof of our main
result based on nonstandard models of arithmetic
A theory of hyperfinite sets
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS,
which is based on the idea of existence of proper subclasses of big finite
sets. We demonstrate how theorems of classical continuous mathematics can be
transfered to THS, prove consistency of THS and present some applications.Comment: 28 page
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