Paradoxes that remain paradoxical even when you understand the underlying theory

Question

It strikes me that the Banach-Tarski paradox (rearranging ball partitions) is not dispelled even when you understand the underlying mathematics. Perhaps Parrondo's paradox in Game Theory (sawtooth losing → winning) is analogous, in that it retains for me a sense of magic even after studying simulations. Perhaps Simpson's paradox in statistics (two plusses's→minus) is borderline, in that, even though it is easy to fall into it, it is also easy to see why one's intuition was incorrect.

Q.Which mathematical "paradoxes" remain paradoxical even when you understand them thoroughly?

I would judge Braess' paradox (adding a shortcut to a road network impedes traffic) as the type of paradox that is dispelled upon understanding it. Whereas the Banach-Tarski paradox remains (for me) paradoxical even though I think I understand the mathematics behind it.

A defensible answer to Q is: No paradoxes remain paradoxical when thoroughly understood—that's what it means to "understand"!

Answer 1

One of my favorites: The real numbers are a vector space over the rationals. Therefore there is a basis for this vector space (a consequence of the Axiom of Choice), and such one basis must lie in the unit interval, since you can replace any basis element by a multiple between 0 and 1.

A lot of paradoxes come from the Axiom of Choice, which nevertheless strikes me as intuitive and a lot of paradoxes come from our failure to understand that infinite sets don't have to correspond to our expectations for finite sets.

Answer 2

From the title it is clear that the issue is psychological. Purely personal example: Though this is not a paradox, existence of countable dense and sigma additivity leads to the notion that there exist an open dense set in the real line of arbitrarily small measure.

Another thing is we accept the standard three axioms a metric has to satisfy. This leads to $p$-adic metric spaces where two open sets can intersect only one is contained in another.

I am reminded of the saying, " A man convinced against his will is of the same opinion still".

Let me quote more. Einstein about statistical/quantum mechanics,"God does not play dice".

Gordon on Hilbert's finiteness theorem on invariants "This is not mathematics, this is theology".

I think Kronecker also had problems with Cantor's theory of infinity.