This is the annotated version of the ten questions about intuitionism:
The function that is 1 for x ≥ 0 and 0 for x < 0 is not total, because it is not defined if you do not know whether x ≥ 0 or x < 0.
If you only get the values of the function to an arbitrary precision, you cannot calculate a zero to an arbitrary precision.
This originates from Brouwer. He has the three infinities "countably infinite", "countably infinite unfinished" and "continuous". The reason that Cantor's theorem does not apply is that the power set does not exist as a finished whole.
The continuum hypothesis is independent of the axiom system ZFC, which might be taken as characterising what sets are.
The continuum hypothesis talks about subsets of the line, and Freiling has shown how to "disprove" it using plausible probabilistic reasoning, so it might be considered meaningful. However, the big cardinal axioms talk about sets that are so big that they are completely irrelevant for all "normal" mathematics. Many people therefore consider the "big cardinal axioms" to be meaningless.
Just build two machines, one that has the yes light on and one that has the no light on. You now will have built the machine that was required, even if you do not know which of the two it is.
In classical propositional logic the formula ( p → q ) ∨ ( q → p ) is a provable statement.
Constructive mathematics is a more precise version of classical mathematics, just like quantum mechanics is a more precise version of classical mechanics.
Different logics are analogous to different kinds of algebraic structures. Some statements hold in Abelian groups but not in non-Abelian groups. Similarly some statements can be proved using classical reasoning but not using constructive reasoning. So it all depends on the context, on the meaning of your words, whether you can prove a theorem or not.
Some logical principles are accepted by more people than other logical principles. So in some sense those are "more true". (The way of phrasing this question is of course a variant of the famous statement from Orwell's Animal Farm.)