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Sunday, September 03, 2006

Weird Things Can Happen with Infinity

Dead Beat is warning you - he is on an infinite roll - you have got to bear with him or settle for the finite - read on at your peril

The symbol for infinity has been around for more than two thousand years. The Romans used it to represent 1000, a BIG number to them. About 1650 the English mathematician, John Wallis, proposed that stand for INFINITY, and that stuck. The concept of infinity has tantalized and sometimes troubled mankind even longer. Zeno of Elea (495 BC?-425 BC?), an early Greek thinker, is remembered for his paradoxes of motion that are rooted in deep questions about the nature of time and space and in some misconceptions about infinity. Most religions attempt to explain in their own ways the mysteries and vagaries of the infinite.

In the early 1600's Galileo began to show signs of a modern attitude toward the infinite, when he proposed that "infinity should obey a different arithmetic than finite numbers." But it was not until the late 19th century that Georg Cantor (1845-1918), a German mathematician, finally put infinity on a firm logical foundation and described a way to do arithmetic with infinite quantities useful to mathematics. His basic definition was simple: a collection is infinite, if some of its parts are as big as the whole. For example, even though from one point of view the entire list of numbers we count with {1,2,3,4,5,.......} is twice as large as the list of even numbers {2,4,6,8,10,.......}, the two lists can be matched-up in a one-to-one fashion.

So the two lists are exactly the same size, infinite. (This idea has been amusingly elaborated in the story of "The Hotel Ad Infinitum" as told by David Stacy.)

Cantor was able to demonstrate that there are different sizes of infinity. The infinity of decimal numbers that are bigger than zero but smaller than one is greater than the infinity of counting numbers. (Click to see Cantor's "diagonalization proof.")

Here's a question that might tax your intuition: There are infinitely many COUNTING NUMBERS and infinitely many FRACTIONS. Which infinity is greater? Think about it, then click for an explanation.

The bottom line is that weird things can happen with infinity.

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