Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible).
The weirdness of the axiom of choice only really comes through when you consider bigger and bigger collections of sets.
For example:
- sets indexed by the natural numbers: S_1, S_2, ...
It seems totally reasonable that you should be able to make a new set by picking something from the first set, something from the second set, etc
- sets indexed by continuous time (i.e. real numbers). Here it's a bit less 'obvious'. If I have sets S_t for _every_ time t > 0, can I really make choices 'fast' enough? What if the sets are so unstructured that I'm forced to stop and look at each set in turn to make my choice?
- sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?
I feel like the last example demonstrates how powerful the full axiom of choice actually is.
NB - I'm a dilettante rather than an actual logician, so there may be mathematical inaccuracies here.
> Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible).
You could form e.g. the non-algebraic reals, which is almost all of the reals.
> sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?
To the extent to which you can form that indexed collection of sets in the first place, surely you can form a similarly indexed collection of elements of them the same way. How can you say you've formed a non-empty set if you can't select an element of it? If we permit ourselves to form this indexed collection "lazily", surely we can do the choice "lazily" as well. (Just my intuition about these things)
The weirdness of the axiom of choice only really comes through when you consider bigger and bigger collections of sets.
For example:
- sets indexed by the natural numbers: S_1, S_2, ... It seems totally reasonable that you should be able to make a new set by picking something from the first set, something from the second set, etc
- sets indexed by continuous time (i.e. real numbers). Here it's a bit less 'obvious'. If I have sets S_t for _every_ time t > 0, can I really make choices 'fast' enough? What if the sets are so unstructured that I'm forced to stop and look at each set in turn to make my choice?
- sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?
I feel like the last example demonstrates how powerful the full axiom of choice actually is.
NB - I'm a dilettante rather than an actual logician, so there may be mathematical inaccuracies here.