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Erik Adolfsson: The axiom of choice

Time: Fri 2021-06-04 11.00 - 12.00

Location: Meeting ID: 662 7911 1082

Respondent: Erik Adolfsson

Abstract

Sets are in a sense the most fundamental mathematical objects. Modern set theory is built up by axioms among which the axiom of choice is often included. Imagine there are urns, each containing different coloured marbles, the axiom of choice says that you are allowed to pick one marble from each urn. Sounds easy enough, you might think, but notice that we did not specify how many urns there are. The power of the axiom of choice is that it holds true even for an infinite amount of urns. 

In mathematics this has some counter-intuitive consequences. For example it implies that all sets can be well ordered, even the real numbers for which we can not currently construct a well ordering. Another consequence is the so called Banach-Tarski paradox which states that you can take a ball, split it up into a number of pieces then rearrange the pieces and put them back together such that you obtain two identicalcopies of the original ball. These concepts along with others will be explored further in the upcoming sections. 

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