Ok, but I'm not sure what your point is. Throwing out only the axiom of choice (which is the difference between ZFC and ZF) does not make things constructive. If you're thinking of Diaconescu's theorem, it's that choice implies the law of the excluded middle, but the converse is certainly not true.

I was objecting to this:

> Generally a lot of counter-intuitive constructions disappear when eliminating the axiom of choice (such as [...] non-continuous functions).