Well not taking the "axiom of choice" for granted is really
really narrowing what can be reasoned about in a really really not
funny way. It makes it for example very "difficult" to invent real
numbers. Well apparently recently some guy published a book which is
basically proposing that the world is just a FSA, so we can see again
that this inconvenience appears to be still very compelling to people
who never had to deal with complicated stuff like for example fluid
dynamics and the associated differential equations :-).
But if talking about actual computers, and since those are in esp.
finite, it may very well be possible to get around without it. ;-)
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