May 8th at 17:30, in ILLC Seminar Room (F1.15)
In this talk, we will construct the set of Hyperreal Numbers using the help of Model Theory. The set of Hyperreal Numbers is a field containing real numbers with the addition of "infinitely small" and "infinitely big" numbers.
We will begin with some historical background of Newton's (and Leibniz's, as well) work (differentiation) and why he needed the concept of "infinitely small" numbers. Then to construct the set of Hyperreal Numbers, we will introduce some Model Theoretic concepts (such as languages, structures, sentences and elementarily equivalence) and Łoś's Theorem. Then, we will construct the nonstandard extension of the set of real numbers which we will call "the set of Hyperreal Numbers" and we will proceed with examples of some actual hyperreal numbers and the extensions of some classical functions from standard analysis, such as exponential function and trigonometric functions. If time permits, we will see some basic theorems in Nonstandard Analysis, such as Robinson's Compactness Criterion.