Sunday, August 1, 2010

The End of Infinity (Infinity of Fun Part III)

by David Siegel Bernstein, PhD

To conclude my series on infinity, I will now attempt to answer the question: does size does matter? Answer: maybe (at least for infinity . . . which is what I am talking about).

According to the mathematician Georg Cantor there is an infinite number of infinities including "little bits" of infinity. Meaning, not all infinities are created equal. No fair! Or is it? Who knows? Remember infinity is an idea not a number.

Consider the -- artfully designed -- Venn diagram below. It represents infinite sized sets nestled inside other infinite sets. The inner most set represents the universe of whole (counting) numbers, but as shown this is only a subset of integers (whole + negative numbers) which is a subset of rational numbers (whole + negative + fractions/decimals), which is a subset of real numbers (all the above + irrational numbers).

Cantor demonstrated that these sets cannot be completely matched up; what he called a lack of 1-1 correspondence. Simply put, if you were to keep using a whole number to cancel out an integer number, once you've exhausted the your stash of whole numbers you would still have integers left over after -- even though you had an infinite number of whole numbers!

By the way, there are sets that lay beyond real numbers called transfinite numbers. Good luck thinking about those! If anyone is interested in learning more about them, let me knew and I'll cover it in a future post.

See you next month in time and space with a new topic.

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