Series: Penn State Logic Seminar
Date: Tuesday, April 4, 2000
Speaker: Stephen G. Simpson (Penn State, Math)
Title: A World Where All Definable Real Numbers Are Computable
Time: 2:30 - 3:20 PM
Place: 219 Thomas Building
Abstract:
An attractive foundational program is ``computable analysis'', i.e.,
the development of mathematics in the computable world. However, it
is also known that the assumption that all real numbers are computable
conflicts with many basic, well known theorems of real analysis, e.g.,
the maximum principle for continuous real-valued functions on a closed
bounded interval. We attempt to strike a compromise between these
conflicting requirements. We do this by exhibiting a world where the
main theorems of real analysis hold, yet all *definable* real numbers
are computable. In technical terms, we construct an omega-model of
WKL_0 in which all definable reals are computable. In fact, for all
reals X and Y, if X is definable from Y then X is computable from Y.
Details are in my paper ``Pi01 Sets and Models of WKL0'' at
http://www.math.psu.edu/simpson/papers/.