Ten Questions about Intuitionism

This is the annotated version of the ten questions about intuitionism:

Do you agree that it is impossible to define a total function from the reals to the reals which is not continuous?
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.
Do you agree that the intermediate value theorem does not hold the way that it is normally stated?
If you only get the values of the function to an arbitrary precision, you cannot calculate a zero to an arbitrary precision.
Do you agree that there are only three infinite cardinalities?
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.
Do you agree that the continuum hypothesis is a meaningful statement that has a definite truth value, even if we do not know what it is?
The continuum hypothesis is independent of the axiom system ZFC, which might be taken as characterising what sets are.
Do you agree that the axiom which states the existence of an inaccessible cardinal is a meaningful statement that has a definite truth value, even if we do not know what it is?
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.
Do you agree that for any mathematical question it is easy to build a machine with two lights, yes and no, where the light marked yes will be on if it is true and the light marked no will be on if it is false?
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.
Do you agree that for any two statements the first implies the second or the second implies the first?
In classical propositional logic the formula ( p → q ) ∨ ( q → p ) is a provable statement.
Do you agree that a constructive proof of a theorem gives more insight than a classical proof?
Constructive mathematics is a more precise version of classical mathematics, just like quantum mechanics is a more precise version of classical mechanics.
Do you agree that mathematics can be done using different kinds of reasoning, and that depending on the situation different kinds of reasoning are appropriate?
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.
Do you agree that all mathematical truths are true, but that some mathematical truths are more true than other mathematical truths?
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.)