I was interpreting "something" as "at least one thing." Almost surely my understanding of mathematics as a whole is incorrect somewhere, but there are a handful of mathematical statements that I believe with complete metaphysical certitude.

What you are really doing is making the claim "Given that what I know about mathematics is correct, then the intersection of a set and its complement is the empty set."

"Correct" is an unclear word, here. Suppose I start off with a handful of axioms. What is the probability that one of the axioms is true / correct? In the context of that system, 1, since it's the starting point. Now, the axioms might not be useful or relevant to reality, and the axioms may conflict and thus the system isn't internally consistent (i.e. statements having probability 0 and 1 simultaneously). And so the geometer who is only 1-epsilon sure that Euclid's axioms describe the real world will be able to update gracefully when presented with evidence that real space is curved, even though they retain the same confidence in their Euclidean proofs (as they apply to abstract concepts).

Basically, I only agree with this post when it comes to statements about which uncertainty is reasonable. If you require 1-epsilon certainty for anything, even P(A|A), then you break the math of probability.