member from all pairs of shoes, e.g. ‘pick the left one’, but that there is no corresponding rule for picking from all pairs of socks.
Mathematicians worried for some time about the axiom of choice – it seems to be an obvious principle to have, but it couldn’t be shown to follow from other axioms, hence remaining an axiom. In the late 1930s, Kurt Gödel showed that it couldn’t be disproved from the other axioms of set theory/logic and then in the 1960s Paul Cohen showed that it couldn’t be proved from those axioms either, meaning it is indeed an independent axiom. The fact that it is needed to prove the existence of a Hamel basis above shows how strange a function it is which satisfies (2) but not (1) – possibly why the FP1 textbook chooses not to worry too much about the question.
References
Torchinsky, A. Real Variables [1988]
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