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.")
