There is always a gap between formalism and intuition and sometimes quite interesting things happen between these gaps. The Banach-Tarski paradox is labeled a p

There is always a gap between formalism and intuition and sometimes quite interesting things happen between these gaps. The Banach-Tarski paradox is labeled a paradox because it says something about how geometry and topology are formalized in set theory. There are other formalizations of geometry and topology where such things are not possible.