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Cordelia Jansson: The Logic behind Kőnig’s Lemma

Time: Mon 2021-05-31 12.00 - 13.00

Location: Meeting ID: 662 7911 1082

Respondent: Cordelia Jansson

Abstract

The goal of this paper is to uncover the complexity of Kőnig’s Lemma and find a deeper understanding of how axioms and theorems are connected. In order to get an insight on the logic behind this lemma, different versions of the lemma are introduced, the relative strengths of them are compared and some applications are discovered. The role of mathematical logic is presented together with the rise of axiomatic systems such as Peano Axioms and two different set theories, namely “Zermelo-Fraenkel set theory” (ZF) and “Zermelo-Fraenkel set theory with Axiom of Choice” (ZFC). A comparison between the two set theories ZF and ZFC is given with a particular focus on one of the most discussed axioms in mathematics, namely the “Axiom of Choice”. Some weaker versions of this axiom are then introduced and compared, in particular “Weak Axiom of choice for countable families of finite sets”.

Mathematical statements equivalent to Weak Kőnig’s Lemma as well as Kőnig’s Lemma are found and the axioms necessary to prove these equivalence relations are investigated.

Finally, the axiomatic systems for which weak Kőnig’s Lemma and Kőnig’s Lemma exist are defined.

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Last changed: May 27, 2021