MENTAL MODEL · M3205

Banach-Tarski Paradox

Learn & MetacognitionHigh supportCognitive Science

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Version 1.0.0 · Updated 2026-07-30

CORE DEFINITION

In three-dimensional Euclidean space, a solid ball can be decomposed into a finite number of pieces and then reassembled via rigid motions into two identical copies of the original ball. This seemingly impossible result relies on the Axiom of Choice and the existence of non-measurable sets.

SCAFFOLDING EFFECT

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Mathematical limit cognition. It reveals the conflict between intuitive geometric intuition and rigorous mathematics, demonstrates the counterintuitive consequences of the Axiom of Choice, and is significant for understanding philosophical issues in the foundations of mathematics.

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In three-dimensional Euclidean space, a solid ball can be decomposed into a finite number of disjoint subsets, which can be reassembled via rigid motions (rotations and translations) into two new balls with the same volume as the original. It relies on the Axiom of Choice and 'non-measurable sets', and does not conserve volume.

MINIMUM ACTION

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Practice this model in one real situation:

As a thought experiment to understand the consequences of 'infinity', 'non-measurable sets', and the Axiom of Choice: 1) Accept non-intuitive conclusions under the axioms of set theory
Distinguish mathematical abstraction from physical realizability
Use it to test the boundaries of intuition regarding 'volume conservation'.

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Source support: Explicit

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en.wikipedia.orghttps://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradoxZH · Explicit
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