by Leonard Wapner (2005). A non-mathematical introduction into set theory and transfinite numbers, culminating in a thorough discussion of the famous Banach-Tarski Paradox. Take a sphere in three or more dimensions and partition it into four non-overlapping subsets. When these are appropriately moved around and re-assembled, you end up with two balls, each with the same volume as the original sphere! No cheating occurs – the subsets are only translated and rotated, no stretching or adding of new matter. This violates our deeply held intuitions about conservation of space/volume. Controversial when it was published by the two Polish mathematicians Banach and Tarski in 1924, this result is now accepted as a consequence of the axiom of choice. The paradox does not apply to `real’ physical space as the subsets in question are non-measurable – they can’t be obtained by cutting the sphere with a knife (but maybe with Philip Pullman’s Subtle Knife from his His Dark Materials). Ultimately, the Banach-Tarski paradox is a further weird consequence of infinite numbers and sets similar to Hilbert’s Hotel with its infinite rooms. Wapner’s book does an admirable job of conveying the relevant mathematics in just over 200 pages.

# The Pea and the Sun: A Mathematical Paradoxon

**19**
*Friday*
Jan 2007

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