The Axiom of Choice

references: wikiwand

The idea of cutting things into pieces and ending up with twice what you started with depends, as previously mentioned, on the Axiom of Choice- so just what is the Axiom of Choice ?

Well, it is the fairly reasonable idea that the set you get from combining two non-empty sets, itself, not empty.

Informally, if you have got a number of bags- even an infinite number of bags- that each contain balls, you can make a selection of exactly one ball from each bag.

If the number of bags is finite, or there’s something that distinguishes the balls, there’s no problem. However, with an infinite number of bags and identical balls, there’s nothing in the other axiom of set theory to say you can do this. In fact, Kurt Gödel proved that the opposite axiom was consistent with the other axioms- meaning you have a free choice about whether to accept the Axiom of Choice.

If the elements in each of the sets in Si are mutually exclusive then it follows that there must exist a set Xi containing one element from each oh the Si sets.

While it’s used uncontroversially by most set theorists, Tarski’s original paper connecting the cardinality (size) of two related infinite sets was rejected by two referees: one, Frechet, because the two things he was relating were obviously true, so it didn’t count as a new result; the other, Lebesgue, because the two things he was relating were obviously false, so the work was of no interest.

‘ The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.’ Bertrand Russell

Thank you for reading.😊

References: Cracking Maths