[KirillPanov]

Spoiler: the cut line between the two apple-halves is fractal in shape, with infinite surface area and taking forever to cut at any finite cutting speed.

It's sort of hilarious to see a physics site mention the Banach-Tarski paradox. It is, after all, the most obvious hole poked in the most basic working assumption used by physicists: that space and time are measured with real numbers.

I've seen physicists go to pretty absurd extremes to avoid thinking about the problems this creates. Fixing it properly is not easy: simply dropping the axiom of choice leaves you unable to do useful physics. Getting back to a useful state, making all sets Lebesgue, can only be done with large cardinals:

https://www.jstor.org/stable/1970696

Large cardinals are pretty exotic even by the standards of mathematicians. In many departments they are in fact the domain of logicians. In fact, the existence of certain classes of Woodin cardinals is equivalent to the Axiom of Determinacy (AD), which is the "mathematically respectable" way of investigating logics with infinitary conjunction/disjunction. In fact, AD is precisely the Law of Excluded Middle (A or not-A) for logics with infinitely-long conjunctions.

Quite odd that something so ethereal would be connected to a tangible act like cutting an apple in half.