Nonterminating decimals do not make sense.

The Banach-Tarski paradox proves how f#cked up the real numbers are. Logical peculiarities confuse our intuitions about “length”, “density”, “volume”, etc. within the continuum (ℝ) of nonterminating decimals. Which is why Measure Theory is a graduate-level mathematics course. These peculiarities were noticed around the turn of the 20th century and perhaps never satisfactorily resolved. (Hence I disagree with the use of real numbers in economic theory: they aren’t what you think they are.)

Axiom of Choice → Garbage

The paradox states that if you assumed the axiom of choice (or Zorn’s Lemma or the well-ordering of ℝ or the trichotomy law), then you could take one ball and make two balls out of it. It follows that you could make seven balls or thirty-seven out of just one. That doesn’t sound like real matter (it’s not; it’s the infinitely infinite mathematical continuum).

I can’t think of anything in real life that that does sound like. Conservation-of-mass-type constraints hold in economics (finite budget), probability (∑pᵢ=1), text mining, and in all the phase and state spaces I can think of as well. Generally you don’t make something out of nothing.

If it’s broke, throw it out.

The logical rule-of-inference Modus Tollens says that if A→B and ¬B, then ¬A. For example if leaving the fridge open overnight leads to rotten food, and the food is not rotten, I conclude that the fridge was not open overnight. Let A = Axiom of Choice and B = Banach-Tarski Paradox. Axiom of Choice leads to Banach Tarski paradox; said paradox is false; so why don’t we reject the Axiom of Choice? I have never gotten a satisfactory answer about that. ℝ is still used as a base corpus in dynamical systems, economics, fuzzy logic, finance, fluid dynamics, and as far as I can tell, everywhere.

How does the proof of paradox work?

The proof gives instructions of how to:

  1. Partition a solid ball into five unmeasurable disjoint subsets.
  2. Move them around (rigidly, without adding mass).
  3. Get a new solid ball, whilst leaving the first ball intact.

The internet has several readable, detailed explanations of the above. You’ll end up reading about Fuchsian groups, Henri Lebesgue’s measure, and hyperbolic geometry (& the Poincaré disk) along the way.

Stan Wagon has also written a Mathematica script to display the subsets in a hyperbolic geometry (whence these pictures come). Thanks, Stan!


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