Aksiomatska nestandardna analiza : delo diplomskega seminarja

Štampar, Kevin (author), Kandić, Marko (mentor)

Cilj diplomskega dela je z uvedbo infinitezemalnih števil predstaviti drugačen pristop h klasični analizi. Infinitezimalna števila uvedemo s pomočjo petih aksiomov. Največji poudarek je na principu prenosa, ki povezuje trditve, ki veljajo nad hiperrealnimi števili, s trditvami, ki veljajo nad realnimi števili. Skozi diplomsko delo dokazujemo, da so definicije iz nestandardne analize ekvivalentne tistim iz klasične analize, nato pa nove definicije uporabimo za dokaze izrekov iz klasične analize. Zaradi obširnosti klasične analize, se v diplomskem delu omejimo na koncepte zveznosti, limite, odvoda in kompaktnosti.

• Language: Slovenian
• Keywords: infinitezimalno število, princip prenosa, standardni del, hiperrealna števila
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-120181
• UDC: 517
• COBISS.SI-ID: 58753027
• Publication date in RUL: 17.09.2020

Secondary language

• Language: English
• Title: Non-standard analysis, axiomatically

Abstract

In the thesis we introduce infinitesimals to present a new approach to classical analysis. We do so by introducing a set of five axioms. The emphasis is on the transfer axiom which helps us transfer theorems from nonstandard analysis to classical analysis. Throughout the thesis we show that the definitions in non-standard analysis are equivalent to those from classical analysis, and use non-standard definitions to prove theorems from classical analysis. Due to the large reach of classical analysis, we limit ourselves to concepts of continuity, limits, differentiation and compactness.

• Keywords: infinitesimal, transfer axiom, standard part, hyperreal number