Topos Theory
The Banach-Tarski Paradox is a theorem in set-theoretic geometry that states it is possible to decompose a solid sphere into a finite number of non-overlapping pieces and then reassemble those pieces into two identical solid spheres, each the same size as the original. This counterintuitive result challenges our understanding of volume and measure in conventional geometry, and is closely related to concepts in independence results and alternative foundations.
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