We can all agree that human intuition is grand but not magical, I hope? Here is my point of view: you are having difficulty teaching a computer to "make that sort of intuitive reasoning" because that sort of reasoning is not quite right.

"That sort of reasoning" is a good heuristic for discovering true facts about the world (for instance, discovering interesting sequences of symbols that constitute a formal proof of the Paris-Harrington theorem), and to that extent it surely can be taught to a computer. But it does not itself express a true fact about the world, and because of that you are limited in your ability to make it part of the premises on which a computer operates (such as the limitation discussed in the OP).

So I've been thinking lately, anyway.