
Independence of the Axiom of Choice (for programmers) - Smaug123
https://www.patrickstevens.co.uk/independence-of-choice/
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fpoling
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)

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Smaug123
Banach-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.

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giomasce
Correct. 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.

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Smaug123
My 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.

