To Infinity and Beyond!
March 17, 2010 6:24 AM Subscribe
Sure, big numbers are fine. But infinity (in the set theoretic sense) is where the fun really starts. Developed almost entirely by one man in the late 19th century, set theory now forms the foundation of modern mathematics. Cantor showed that not all infinite sets are the same size. Notably, while there are just as many integers as rational numbers, there are more real numbers than integers. These results, along with others that soon followed like the axiom of choice, led to several fascinating consequences:
- The Cantor Set is a fractal constructed by repeatedly removing the middle third of a line segment. You end up with a set that has no length, yet as many points as the reals.
- The Hilbert Hotel was used by David Hilbert to demonstrate how an infinite hotel could be completely full yet still accommodate infinitely more guests. (Another retelling of Hilbert's Hotel.)
- The continuum hypothesis, which deals with whether there's a set that has a size between that of the integers and reals, has been shown to be independent of the basic mathematical model. Both it and its negation are consistent within the standard logical framework math is built on.
- The Banach-Tarski Paradox showed that it's possible to cut a sphere into a finite number of pieces and put the pieces back together again so that you end up with two solid spheres of the same size as the original!
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