Monday, August 15, 2005

Neo-Aristotelian? Stupidity

     This link to the Math Academy is official math, and it is a pretty easy read about infinity.  It includes a good history, which explains how Aristotle didn't accept infinity, and the eventual acceptance of it took a long time, and made Georg Cantor a controversial figure.  Galileo helped start the infinity ball rolling with this, called Galileo's Paradox

Galileo (b. 1564) had given the matter [infinity] much thought, and noticed the following curious fact: if you take the set of natural numbers and remove exactly half of them, the remainder is as large a set as it was before. This can be seen, for example, by removing all the odd numbers from the set, so that only the even numbers remain. By then pairing every natural number n with the even number 2n, we see that the set of even numbers is equinumerous with the set of all natural numbers. Galileo had hit upon the very principle by which mathematicians in our day actually define the notion of infinite set, but to him it was too outlandish a result to warrant further study. He considered it a paradox, and “Galileo's Paradox” it has been called ever since.

     I simply can't accept this.  I recognize the existence of infinity, and that there are infinite rational numbers between any other two rational numbers, but I don't want to accept that there is a 1-to-1 mapping between the even integers and the integers.

     For any integer x greater than 2, the number of whole integers less than x is always twice the number of evens, or twice that number plus one.  This is not a limit sequence.  The number of "evens" is always slightly less than half, or exactly half, of the number of integers less than any real number.

     If I was right, at least some definitions of infinity would have to be changed.  I simply think that it should be possible for sets of infinite cardinality to have non-equal cardinality. 

     Back in college, I was OK at math.  I scored in the top 25% of the annual Putnam exam one year, one of 1989, 1990 or 1991.  A combination of factors (J Glick being better at math than I, combined with limited occupational options for professional mathematicians) caused me to switch my emphasis to economics.  I figured it was halfways between math and art.

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