Monday, September 01, 2008

Banach-Tarski

Following on from my last post, I’ve noticed that those introducing standard set theory often say it’s a good foundation for maths because it gave answers to questions that mathematicians were asking, a hundred years ago (even if it can’t show them to be the correct ones), e.g. the problem of integrating a curve that takes the value of 1 for irrationals between 0 and 1, and otherwise takes the value 0, to which standard measure theory gives the answer of 1. But note that such problems only arise within this kind of foundation. If continua are absolutely continuous (e.g. they contain 1/0 points) and simple infinities are indefinitely extensible, for example, then no such problems could arise because only a finite number of exceptional points could be given.
......Furthermore standard set theory throws up questions about measures that it certainly cannot answer. The pieces of the decomposed sphere of the Banach-Tarski paradox, for example, cannot be given any measures consistently—not even measures of 0. That paradox is usually 'resolved' by something like the assertion that it is really just an aspect of the proof of a theorem that certain sets of points cannot be given certain kinds of measure. (That assertion is usually followed by a question about why one would expect all sets of points to have measures, in view of such examples of counter-intuitive behaviour with infinite sets as Hilbert’s Hotel.) It’s usually been forgotten, by that stage of the exposition, why set theory was being entertained in the first place.

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