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Free groups and the Axiom of Choice

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Mathematical, Foundational and Computational Aspects of the Higher Infinite

The role of the Axiom of Choice in Mathematics has been studied extensively. Given a theorem of ZFC , one may ask how strong it is compared to the Axiom of Choice. Although a large collection of results has been analysed in this way, there are still simple and elegant theorems that offer resistance. One such result is the Nielsen-Schreier theorem, which states that subgroups of free groups are free.

I will introduce recent results that help to establish the strength of Nielsen-Schreier, focussing on the method of representative functions. Then I discuss potential applications of this technique to other algebraic structures admitting a basis, such as free abelian groups and vector spaces.

This talk is part of the Isaac Newton Institute Seminar Series series.

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