Milton Ernarp: Some Basic Results in Non-Standard Analysis

Abstract.

In this thesis, we lay the groundwork for non-standard analysis and derive some classic results from real analysis, through alternative methods. Non-standard analysis extends the real number line to the hyperreal number line, which we construct by introducing ultrafilters. The hyperreal number-line includes infinitesimal numbers, whose absolute value is less than any positive real number, and unlimited numbers, whose absolute value is greater than any real number. A fundamental result we prove is the transfer principle, which allows first-order statements about one structure, to be transferred to a corresponding statement about the other structure. Using transfer, we establish a basic theory of non-standard analysis, including proofs of classic theorems such as the Squeeze Theorem, the Intermediate Value Theorem, the Chain Rule, and the Fundamental Theorem of Calculus. We do this using infinitesimals — akin to the original methods of Leibniz — instead of the standard limit-based formulations. The thesis shows how non-standard methods can offer a more intuitive alternative for analysis.