i wrote this a couple of decades ago :P

if you have 32 numbers in a set, each number must have at least enough information in it to distinguish it from 32 other numbers
in the simplest case, bitwise you could say that the values are
[['0', '0', '0', '0', '0'], ['0', '0', '0', '0', '1'], ['0', '0', '0', '1', '0'],
 ['0', '0', '0', '1', '1'], ['0', '0', '1', '0', '0'], ['0', '0', '1', '0', '1'],
 ['0', '0', '1', '1', '0'], ['0', '0', '1', '1', '1'], ['0', '1', '0', '0', '0'],
 ['0', '1', '0', '0', '1'], ['0', '1', '0', '1', '0'], ['0', '1', '0', '1', '1'],
 ['0', '1', '1', '0', '0'], ['0', '1', '1', '0', '1'], ['0', '1', '1', '1','0'],
['0', '1', '1', '1', '1'], ['1', '0', '0', '0', '0'], ['1', '0', '0', '0', '1'],
['1', '0', '0', '1', '0'], ['1', '0', '0', '1', '1'], ['1', '0', '1', '0', '0'],
['1', '0', '1', '0', '1'], ['1', '0', '1', '1', '0'], ['1', '0', '1', '1', '1'],
['1', '1', '0', '0', '0'], ['1', '1', '0', '0', '1'], ['1', '1', '0', '1', '0'],
 ['1', '1', '0', '1', '1'], ['1', '1', '1', '0', '0'], ['1', '1', '1','0', '1'],
 ['1', '1', '1', '1', '0'], ['1', '1', '1', '1', '1']]
(this is a set of sequences.)
of course, it's not that simple, because 0 could be represented bitwise by {'0'}, or even {}.
but in that case, how does {} have enough information in it to distinguish it?
well, the answer is that a set must have a length. the length of that set might be specified in binary by 000.
you need three 0's, not just one, because if you specify length as 0, you need another sequence to say how many
bits you're using to specify your length, and a sequence to say how many bits you're using to store that value, ad infinitum.
but still, how do 3 bits, which is the only information attached to that element, which give it a binary range of 0-7, enough to distinguisg it from 31 other entities? the answer is that 000 is a selector, which selects for the subset of elements of length 0, and {} selects the element from that set, which is only itself.
001 is a selector which selects the subset of elements of length 1, and {'0'} or {'1'} selects from the two elements of that list.
so the set can be thought of, in this conceptualization, as a tree. the lower your node depth, the less information you need to select it.
note that for this argument to work, i think the sequence-length echelon and the number-information echelon both need to be trees, so that 000 represents the path 0, 0, 0 down the first echelon.
you could say that you have 32 unique symbols, so there are no tree paths, but the uniqueness of a unique symbol itself implies information to distinguish it from other symbols, even if that information is no more than the contours of its visual representation.
of course if you have a grab-bag that renders an infinite series of unique symbols with which to fill a set, then we have a problem..
this may be where the axiom of choice comes in, but notice that with such a set, the elements involved are meaningless to us. we cannot treat them as numbers; numbers exist as sequences of specified lengths containing values from a fixed-size set.
to be perfectly fair, if the uniqueness of a symbol is defined by its visual representation, then the specific length of a sequence can also be defined by its overall visual representation. we say the sequence 'ends' when we see a }. but it's still a tree path. e.g. each digit is a branch, and } represents the end of a path.
alternatively, using only symbols, [['1', '1', '1', '1', '0'], ['1', '1', '1', '1', '1']] becomes '[','[','1', '1', '1', '1', '0',']', '[','1', '1', '1', '1', '1',']',']', with some question as to whether the first and last in the sequence can actually be symbols, instead of meta-symbols as in ['[','1', '1', '1', '1', '0',']', ',', '[','1', '1', '1', '1', '1',']']

now take, instead of 32 sequences, an infinite number of them.
for each one to have

the axiom of choice at best is an arbitrary choice that may have interesting theoretical results but no practical applications. and at worse it's a ridiculous option but people won't really know why until somebody sits down and proves it.
the only application for the axiom of choice is that there are some conjectures you can make in math that are meaningless without it and with it can be proven to be true. but since it's only the axiom of choice that made them meaningful, and the axiom of choice isn't itself useful for anything but proving such conjectures, proving those conjectures has no practical benefit.
of course, since the conjecture already exists before it's proven or not, if there's any practical benefit to it then it can be applied to get end results whether or not it's proven.

invoking the axiom of choice is one or more of the following:
--selecting an element that never actually exists, as it has no distinguishing qualities, can't be constructed and hence used, and so on, thus actually adding nothing new into the mix.
--imagining that you are selecting an element by filling out the first few symbols of it with random values in your head, without actually continuing to construct it to the end which is impossible thus making your fantasy meaningless.
--simply an unexamined and unspecific token of nothing but, at best, a restatement of the principles we use to construct numbers or sets. of course, its being unexamined, it's just as likely that it's nothing if not a nondescript token of a general *absconding* from the way we construct, use and think about numbers and sets and infinity. the latter, of course, is more consistent with the observation that the axiom of choice defies constructivist maths.