Famously, the continuum hypothesis, the axiom of choice, and a set of other potentially optional improvements of Zermelo-Frankel Set Theory are potentially true, but unprovable within the formalism of the theory itself.
In such cases, effective proof is absolutely impossible for either the statement itself or its negation. Since one side or the other is classically true, we have candidates for true-but-unprovable results, but, without any agreed criterion for truth beyond language, we can only narrow them down to opposing pairs, and never pick one that is definitely true and unprovable.
These, especially the axiom of choice, seem intuitive to some people, but can be used to produce counterintuitive results, so others feel we have not found the correct intuitive interpretation of them. And still others find other things intuitive, like the determinacy of matrix games, which would contradict the axiom itself. Since we cannot even agree on a criterion for what is intuitive, we cannot possibly imagine that this is a set, much less a recursively enumerable one, because the element operator is ambiguous.
In his Intuitionism, Brouwer sets the bar quite high on what is intuitive. Heyting recasts some of Brouwer's 'basic intuitions' in the form of sets of axioms, and his success here indicates the remainder can probably also be captured that way, though, to some degree, putting them that way defies the intention of the program.
From there we only accept what can be constructed. That means the true statements are recursively enumerable, because only stated axioms and the resulting proven statements are allowed to meet the criteria for intuitiveness.
So by at least one school that attends closely to intuition, you could consider the intuitive propositions recursively enumerable.