To which extent constructive mathematics can recover classical mathematics? Is there classical theorem about computable objects and operations which can not be proven constructively?
I know that many classical theorems can be proven constructively at least after modification. For example intermediate value theorem in analysis. And there is still some important theorem like Falting’s theorem which lacks of proven effective bounds.
My intuition is that when we restrict to computable (in some sense) objects and computable operations on them, we should be able to give a constructive proof when a classical one is valid.
For example, if by computable proposition, we mean decidable one, then we can in fact compute its truth value, so in this sense we have modified version of law of excluded middle. Let $P$ be a decidable predicate on natural numbers, then we can prove countable choice for $P$ constructively.
Now it comes to the question. Is there any counter example?
Note: The modification should have a computable meaning. For example, only adding LEM as hypothesis is not my intention. I expect one can compute the truth value of proposition $p$ before assuming LEM for $p$.
