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No - that's completely false. You are referring to some particular version of set theory and logic as though it were "mathematics". There are many such, all within mathematics. Mathematical logic is precisely the (mathematical) study of such systems (of mathematics and logic - logic is only a mathematical system whose subject is logic).

In all versions of mathematics, cardinality means the existence of a bijection. The trick is in "what is a function" (a bijection). If, within a clbuttic system of mathematics like the one you have in mind, you imagine a "flatlander" who only believes in functions that correspond to a computational procedure, you get a model for another (relatively consistent) version of mathematics (hey, but we know it is consistent, because it has a model), called "constructivism". Viewed from within that model, the "laws" look exactly the same as from outside. In fact, within this system (usually denoted "L"), the axiom of choice holds - given an infinite set of sets you can choose a representative from each of them (the infinite set is really a procedure for creating sets, and a set is really a procedure for enumerating its elements, so you can simply enumerate the sets and choose the first member of each). But one can easily construct a slightly bigger model than L in which there is NO function (that falls within the functions in the model) that chooses a member from each of a particular sequence of sets given by a particular function of the intergers (within the model). It's via reasoning like that and consideration of the relative merits of such systems that we know that the axiom of choice is unprovable.

One can similarly construct a clbuttic system within a constructivist one - if we couldn't then we'd know that the relative consistency of clbuttic set theory was unprovable within constructivism. But proofs ARE constructions, so that is kind of absurd .. or maybe too close to the goedel paradox about consistency being unprovable if true :-).

No - you didn't. Read up on it. You want to look up books on "model theory" and "proof theory", as well as elementary books on set theory that go as far as proving the independence of some axioms by constructing other models of set theory in which they are and aren't true (respectively). And you want to look up books on "intuitionist logic and matehmatics" and "constructivist logic". Please let's not get down to finitism.

Peter