Thursday, January 23, 2020
Infinity in a Nutshell :: Mathematics Math
Infinity in a Nutshell Infinity has long been an idea surrounded with mystery and confusion. Aristotle ridiculed the idea, Galileo threw aside in disgust, and Newton tried to step-side the issue completely. However, Georg Cantor changed what mathematicians thought about infinity in a series of radical ideas. While you really should read my full report if you want to learn about infinity, this paper is simply gets your toes wet in CantorÃ¢â¬â¢s concepts. Cantor used very simple proofs to demonstrate ideas such as that there are infinities whose values are greater than other infinities. He also proved there are an infinite number of infinities. While all these ideas take a while to explain, I will go over how Cantor proved that the infinity for real numbers is greater than the infinity for natural numbers. The first important concept to learn, however, is one-to-one correspondence. Since it is impossible to count all the values in an infinite set, Cantor matched numbers in one set to a value in another set. The one set with values still left over was the greater set. To make this explanation more comprehendible, I will use barrels of apples and oranges as an example. Rather then needing to count, simply take one apple from a barrel and one orange from the other barrel and pair them up. Then, put them aside in a separate pile. Repeat this process until one is unable to pair an apple with an orange since there are no more oranges or vice versa. One could then conclude whether he has more apples or oranges without having to count a thing. (Izumi, 2)(Yes, itÃ¢â¬â¢s a bit egotistical to quote myselfÃ¢â¬ ¦) Cantor used what is now known as the diagonalization argument. Making use of proof by contradiction, Cantor assumes all real numbers can correspond with natural numbers. 1 Ã¢â -----Ã¢â â .4 5 7 1 9 4 6 3Ã¢â¬ ¦ 2 Ã¢â -----Ã¢â â .7 2 9 3 8 1 8 9Ã¢â¬ ¦ 3 Ã¢â -----Ã¢â â .3 9 1 6 2 9 2 0Ã¢â¬ ¦ 4 Ã¢â -----Ã¢â â .0 0 0 0 0 6 7 0Ã¢â¬ ¦ (Continued on next page) 5 Ã¢â -----Ã¢â â .9 9 9 9 9 9 9 1Ã¢â¬ ¦ 6 Ã¢â -----Ã¢â â .3 9 3 6 4 6 4 6Ã¢â¬ ¦ Ã¢â¬ ¦ Ã¢â¬ ¦ Cantor created M, where M is a real number that does not correspond with any natural number. Taking the first digit in the first real number, write down any other number for the tenthÃ¢â¬â¢s place of M. Then, take the second digit for the second real number and write down any other number for the hundredthÃ¢â¬â¢s place of M.