Can mathematics turn one ball into two? (The Banach–Tarski paradox)
The mathematical theorem uses a finite decomposition into highly nonmeasurable sets that can be rearranged into two balls of the original size. Its pieces are not chunks a physical cutting machine can isolate, and the theorem is not a recipe for manufacturing matter. OPH emphasizes the same distinction between a mathematical description and an admissible observer operation. A proposed physical transformation needs actual states, access, energy accounting and records of its implementation. Treating arbitrary continuum subsets as available material pieces bypasses those requirements. The result is compatible with ordinary conservation laws because its assumptions concern abstract sets and ideal motions, not a finite physical protocol that doubles a gold sphere.
See also Do infinities exist in reality or only in our descriptions?, Does describing a universe make it exist?
Sources: The Banach–Tarski Paradox: mathematical construction
Also in Chapter 18: Why Does Anything Have Energy?, Chapter 35: What Is Anything Made Of?.
This answer comes from OPH, Observer Patch Holography. OPH is a rigorous, consistent and complete description of the unified, self-creating computational Universe we inhabit. Its consistency is machine-checked: the fixed point at its core is constructed explicitly, and a theory with a model contains no contradiction.
To understand OPH and check every step yourself, start with the free book Reverse Engineering Reality. It builds the theory from the ground up, and the OPH Notebook answers your questions while you read.
read the free bookask the notebookor ask sage:read chapter 13: What Is One Piece of Reality Actually Made Of?watch a universe make its own particles