Forgive me for not knowing much about constructivism. You mention having “a positive reason to believe such a set exists,” but what more reason could there be o

Forgive me for not knowing much about constructivism. You mention having “a positive reason to believe such a set exists,” but what more reason could there be other than the fact that the axioms you’re using imply that such a set exists? Do constructivists believe that a prime with a googol base-10 digits exists in ZF, even though no one has constructed one?