Infinity is a big concept in higher mathematics. From the moment that calculus allows you to “approach” infinity, questions begin to arise about the nature of the infinite. Some are metaphysical, asking about the “realness” of infinity, and others are purely mathematics, asking about the “bigness” of infinity.
Cantor’s Surprising Result
Georg Cantor was a man unafraid of infinity. While most mathematicians were content to work only in finite areas, Cantor took on infinity head first and found some interesting properties. Specifically, he found that there are multiple infinities, and some are strictly bigger than others. He called the smallest infinity countable, and bigger ones became uncountable.
Bigger infinities seem like the subject of a child’s argument about infinity, by saying “infinity + 1.” Of course, infinity + 1 is still just infinity, so that won’t do it. Let’s start by talking about “countable infinity,” or the infinity that is easiest to see. Countable infinity is the size of anything that can be counted. Being counted means that you can line them up in such a way that you can number them (i.e. 1,2,3…) and be sure you didn’t miss any.
Obviously, the counting numbers are countable. The integers are, too. While it seems that there should be more integers, you can count them by inserting the negative numbers into the list in every other spot. That would count them as “1, -1, 2, -2, 3, -3…” and you would hit them all.
It gets weirder with rational numbers. There are infinite rational numbers just between 0 and 1, but they are countable, too. You count them in a weird, diagonal way that I won’t go into here, but trust me, they can be counted. Our first “uncountable” infinity comes up when you get to irrational numbers.
Uncountable infinity can’t be numbered. When you try to list them, you can always find a number you missed. Cantor offered a (relatively) simple proof that the real numbers, when you include the irrational numbers, are uncountable. His proof works as follows:
Suppose you had a numbering scheme that should count all the real numbers. He would go down that list, and for the nth term, read off the nth digit. He would then change it to something different and use it on his new number. Since his “newly constructed number” is different from every other number on your list (at the nth digit at the very least), then you must have missed it in your enumeration. Since you can’t enumerate the list in any way, it must be too big, thus, uncountable.
Accepting Uncountable Infinity
Many people object to uncountable infinity. It isn’t a natural idea in any way. It simply doesn’t make any sense. Some people even object to any infinity at all. Since the universe is finitely bounded according to modern physics, they argue that infinity is a thought construct and not a reality.
They may be right, but these objections are philosophical, not mathematical. Mathematically speaking, these “multiple infinities” are well accepted. In fact, there are an infinite number of these infinities, each larger than the last. These infinities stretch the imagination to consider the properties of our universe, math, and the way that they interact.
Wolfram Math World: Uncountably Infinte
Wolfram Math World: Countably Infinte
Maa.org: Infinity and Intuition
Mattababy.org: Appendix: Infinity