[nuncanada]

I must the only crazy guy that thinks Boltzmann and Weierstrass got the wrong axiomatization and Real Number Set is an oxymoron? They are completely physically unrealizable.

Every number that is not computable in the the Real Set is also not possible to be written down, don't have an algorithm for it, we can't even talk about or name any of them. All we can talk about is this uncountable part of the Real Set as a Set but never about one of the numbers itself.

[pron]

It's not like this issue hasn't come up, you know. That was at the core of the controversy between Hilbert and Brouwer. But if you think that math must be constructive you must realize that the ramifications are significant and the resulting math is very different and has its own counter-intuitive results. For one, analysis becomes much harder (at least so far) and very different; for example, in intuitionistic math, every function on the reals is continuous. Some counter-intuitive results are that a countable set can have an uncountable subset; in fact, not all subsets of a finite set are even finite. Andrej Bauer wrote a nice, mostly readable introduction to constructive math, which you may find interesting: http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2016...

There are numerous philosophical justifications for non-constructive math, but the pragmatic one is that it is -- as far as we know -- consistent, and coincides with constructive math in all physically realizable instances. Hilbert, for example, said that math has "real propositions" (with physically observable consequences), as well as formalistic propositions, which we can accept as they agree with the real propositions in all instances.