On Thursday, January 9, we will have a LIRa session with Imme van den Berg. Everyone is cordially invited!
Speaker: Imme van den Berg, (Department of Mathematics, University of Evora)
Title: A model for orders of magnitude within Nonstandard Analysis
Date and Time: Thursday, January 9 2014, 15:30-17:30
Venue: Science Park 107, Room F1.15
Abstract:
Many arguments deal informally with orders of magnitude of numbers. The aim of this talk is two-fold, to establish an acceptable axiomatics for orders of magnitude and indicate structures which satisfy the axioms or parts of them. Our formalization tries to maintain the intrinsic vagueness: orders of magnitude should be bounded, but stable under at least some additions.
Due to the Archimedean property and Dedekind completion, such a formalization cannot be done with ordinary real numbers, still there is the functional approach through O’s and o’s, and more generally Van der Corput’s neutrices, both have some operational shortcomings.
Nonstandard Analysis disposes of a natural example of order of magnitude: the (external) set of infinitesimals is bounded and closed under addition. Adopting the terminology of Van der Corput, we call a neutrix a additive convex subgroup of the nonstandard reals. An external number is the set-theoretic sum of a nonstandard real and a neutrix.
The external numbers capture the imprecise boundaries of informal orders of magnitude and permit algebraic operations which go beyond the calculus of the O’s and o’s, allowing for total order with a generalized form of Dedekind completion.
In addition we discuss some applications and foundational problems.