Logic Seminar
Evgeny Gordon
Eastern Illinois University
Continuous vs Discrete via Nonstandard Analysis
Abstract: In the second half of the last century a new point of view on interrelation
between the continuous and discrete mathematics started to become popular
among applied mathematicians. According to it the continuous mathematics is
an approximation of the discrete one, but not vice versa. The reason of
this popularity is the widespread use of computers in both applied and
theoretical research. However, the formalization of mathematics based on
this point of view meets serious difficulties in the framework of Cantor's Set
Theory, because we need to deal with not well defined collections, like very big
numbers, or numbers far enough of boundaries of computer memory,
that depend on concrete problems or points of views.
Maybe, the difficulties in mathematically rigorous justification of
theoretical physics have the same reason the axiom of least upper bound is
too strong idealization for physics. A new axiomatic system
(NNST — Naive Nonstandard Set Theory) based on ideas of A.
Robinson's Nonstandard Analysis and P. Vopenka's Alternative Set Theory
will be presented in this talk. The idea of approximation of discrete
structures by continuous ones is implemented in this theory as follows.
Continuous structures appear from finite very big finite ones as
factorizations of accessible substructures of these finite structures
by some indiscirnability relations. The properties in italic here are
not well defined ones. We discuss some theorems formulated and proved in
the framework of NNST related to computer simulations of continuous
structures, which have clear intuitive sense, can be monitored in
computer experiments, but whose formulations in the framework of
Cantor's Set Theory are irrelevant, if not to say unreadable.
Tuesday November 10, 2015 at 4:00 PM in SEO 427