>That's what logic does to intuition. If you can't prove your intuition false, that's generally when you feel like you "know" a thing, or have "proven" a thing to be true.

yes and no.

the very concept of "proving something false" implies a number of assumptions.
the existence of a subject, a critical thought, an external reality that follows decipherable rules and patterns, a language bearing meaning, the existence of the very intuition I am about to disprove.
I would say that these 'core intuitions' cannot be refuted by any rational proof, since rational proof itself presupposes them.