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Mathoverflow has some good discussion of Banach-Tarski [0]. The top-rated posts argue that - The 'problem' with B-T may be related to our notion of "space" (i.

Mathoverflow has some good discussion of Banach-Tarski [0]. The top-rated posts argue that - The 'problem' with B-T may be related to our notion of "space" (i.e. point-set topology) rather than any issues with the axiom of choice - Most practical uses of the axiom of choice could get by with the axiom of countable choice, under which B-T doesn't hold - Mathematics need not directly model the 'real' world or match our physical intuition [0] https://mathoverflow.net/questions/260057/axiom-of-choice-banach-tarski-and-reality https://mathoverflow.net/questions/260057/axiom-of-choice-ba...

dkural 4y ago
Thank you for the summary and the link. I'd also add that our physical intuition also need not match the "real world", it's a model "good enough" for evolutionary purposes that fails in myriad ways.
msla 4y ago
> Mathematics need not directly model the 'real' world or match our physical intuition There's so many other ways to demonstrate this. For example, it's possible to map the integers to the rational numbers one-to-one, which is a good definition for "as many as" when dealing with infinite sets, even though there are infinitely many rational numbers between any two rational numbers. Moreover, there are as many integer multiples of a trillion as there are integers total. This is to do with set cardinality: https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Cardinality ... which again comes back to the axiom of choice: > If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions: > * Any set X with cardinality less than that of the natural numbers [...] is said to be a finite set. > * Any set X that has the same cardinality as the set of the natural numbers [...] is said to be a countably infinite set.[10] > * Any set X with cardinality greater than that of the natural numbers [...] is said to be uncountable. (... with some mathematical notation that doesn't work here elided)

dkural 4y ago

Thank you for the summary and the link. I'd also add that our physical intuition also need not match the "real world", it's a model "good enough" for evolutionary purposes that fails in myriad ways.

msla 4y ago

> Mathematics need not directly model the 'real' world or match our physical intuition There's so many other ways to demonstrate this. For example, it's possible to map the integers to the rational numbers one-to-one, which is a good definition for "as many as" when dealing with infinite sets, even though there are infinitely many rational numbers between any two rational numbers. Moreover, there are as many integer multiples of a trillion as there are integers total. This is to do with set cardinality: https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Cardinality ... which again comes back to the axiom of choice: > If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions: > * Any set X with cardinality less than that of the natural numbers [...] is said to be a finite set. > * Any set X that has the same cardinality as the set of the natural numbers [...] is said to be a countably infinite set.[10] > * Any set X with cardinality greater than that of the natural numbers [...] is said to be uncountable. (... with some mathematical notation that doesn't work here elided)