[IngoBlechschmid]

In constructive mathematics, we don't reject "all infinite constructions". The only axiom which we don't generally assume is the axiom which says "any statement is either true or not true". (Note that we also do not use the counterfactual axiom "there is a statement which is neither true nor false". In fact, we're just agnostic on some truth values.)

In constructive mathematics, there is a perfectly well-defined set of real numbers. The usual diagonalization proof that this set is not a countable set applies.

[upquark]

I said "and other groups that reject all infinite constructions". Some schools of thought within that general intuitionist/constructivist/etc branch of mathematical logic do reject all infinite constructions: https://en.wikipedia.org/wiki/Finitism

Either way, my point above was that this entire branch is not "mainstream math" by any means, AFAIK

[IngoBlechschmid]

I totally agree that mainstream mathematics doesn't have any problem whatsoever with infinite constructions and in fact embraces them.

I just wanted to clarify that intuitionistic and constructive mathematics don't have any problems with infinite constructions either. Finitism and ultrafinitism do, but they're not what's usually called "constructive mathematics".

There are at least three orthogonal axes which you can classify mathematical schools of thought in:

* Is the law of excluded middle accepted? ("Any statement is either true or not true.")

* Are infinite sets accepted? (They are not in finitism, but they are in constructive mathematics and of course in ordinary mathematics.)

* Can constructions implicitly refer to the result of what is being constructed? Is the powerclass of a set again a set? (Yes in ordinary mathematics and in constructive mathematics, no in predicative mathematics.)