On the intuitionism page at Stanford Encyclopedia of Philosophy (SEP), it's said in Section 3.3 that
Because of the finiteness of a natural number in contrast to, for example, a real number, many arithmetical statements of a finite nature that are true in classical mathematics are so in intuitionism as well. For example, in intuitionism every natural number has a prime factorization; there exist computably enumerable sets that are not computable...
This uncomputability result must necessarily use the halting problem, whose uncomputability is proved unconstructively, as it relies on the law of the excluded middle: TM $M$ of input $\langle M\rangle$ either halts or not. This is not constructive because it might be impossible to know whether the TM runs forever. If you can prove for every case that the TM runs forever when it indeed does, then the collection of such proofs results in an algorithm that decides haltingness. So this proof is not constructive. But SEP says that intuitionists accept it! Isn't intuitionism a constructive theory? Or are there other constructive proofs of the halting problem that I don't know?
