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The thing about Axiom of Choice is that combining it with ZF set theory essentially describe a world with magic [1] allowing to turn any object into another one
by fpoling 11y ago
The thing about Axiom of Choice is that combining it with ZF set theory essentially describe a world with magic [1] allowing to turn any object into another one of arbitrary size just by splitting the object into finite number of parts and reassembling them.
So if one belive in magic, then one should consider ZF+AC as a realistic model of our world.
[1] https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
- Smaug123 11y agoBanach-Tarski is a good argument for not believing that the world is modelled by R^n for any n, rather than a good argument for disbelieving the Axiom of Choice. After all, the real world has the Planck length.
- giomasce 11y agoCorrect. ZF is anything but realistic. It is just nice for creating mathematical theories, which can indeed be wonderful and also very useful for our world.
- Smaug123 11y agoMy experience is limited, but I consider that to be why the Continuum Hypothesis isn't very interesting. The continuum is how we try and embed "space" into ZFC, but the way we embed has certain freedoms in it: one of which being the size of the set that is used to represent the continuum. It's the same kind of uninteresting as the difference between the two implementations {x, {x, y}} and {{x}, {x, y}} of the ordered pair.
- fpoling 11y agoBanach-Tarski is a good argument not to rely on AC and stay within, for example, constructive analyzes if one wants however small but a notion of practicality in the results.