A Bit About Measuring Infinity, And The (Infinitely) Infinite Flop
Monday July 31, 2006 by Noam Samuel
It’s hard to measure infinities. On one hand, it seems reasonable that the set of real numbers is larger than the set of natural numbers, but how would you define “larger infinity”?
One of the definitions of a set having more elements (or, as we mathies would say, a higher cardinality) than another is that there cannot exist a mapping of all the elements in the latter set into the former such that every element in the former set has an element in the latter set mapped to it, while there can be such a mapping from the former set to the latter.
For example, it is possible to say that the set of all real numbers is larger (or, more accurately, has a higher cardinality) than the set of natural numbers.
In fact, one of the categorizations of infinity is the differentiation between “countably” and “uncountably” infinite sets. Sets that are countably infinite must be mappable in a one-to-one mapping onto the set of natural numbers, whereas sets that are uncountably infinite must not be. The set of real numbers, for example is uncountably infinite, whereas the set of numbers divisible by 2 is countably infinite.
This, however, does not seem to give a full answer to the intriguing question of how one goes about measuring an infinity. Some infinities seem “larger” than others, but there is no known way to formulate that difference (oftentimes because it is provably not there). And how does one go about saying which set is more infinite than another, and how many infinities are equal to this or that? Well, this is where the second part of my title, the (infinitely) infinite flop comes in.
Despite its name, it is not all that horrible of a flop. The actual reason it is called the (infinitely) infinite flop, is because it is a flop about the subject of things that are (infinitely) infinite.
The idea came when I noticed that lots of infinite sets were infinitely as other infinite sets, meaning that per each elements in the latter set, there is an infinite amount of elements in the former. I came eventually to think that this property might be used to stratify infinite sets.
Thus, I formulated the following definition: Set P is infinite to set Q when there exists a function [f(x)] whose domain is Q and whose range is the set of the infinite subsets of P such that for every two elements a and b in Q, f(a) and f(b) are disjoint. For many practical purposes, however, the definition could just read as the following: “Set P is infinite to set Q if every element in Q maps to an infinite number of unique elements in set P.”
I even created a symbol for this relation: 
. In fact, one of the first complaints I got when I presented my idea to my friends was that the symbol was symmetric, and does thus not properly represent the asymmetry between P and Q. This, however, turned out not to be as much of a problem as we thought.
You see, it starts from the fact that such a stratifying (or, at the very least, positioning) relation cannot, in any way, be reflexive. If it turned out that 
for any P, the whole purpose of constructing the definition goes down the drain.
It didn’t take me long to realize this, and I was soon off to try to prove and disprove the existence of such a “reflexive P”. The task, however, was not easy, and after a few weeks of not being able to do either, I abandoned the idea altogether.
But some time after the creation of The Math Underground, Patrick and I were pining for ideas about which we could write, and I remembered my definition of infiniteness. Without the missing proof, however, I could publish the article just to find out later that the whole idea was useless.
Eventually, we decided to leave it as an open question for the reader, and then went into the nuances of the symbols I used and other parts of the definition when suddenly it hit me.
First of all, there’s an infinite number of prime numbers (there’s proof of that, but I’m going to “chicken out” again since I’m too lazy to write it), each of which has an infinite number of positive integer powers. Now, the set of the positive integer powers of one prime must be disjoint set of positive integer powers of any other prime due to a little thing called the fundamental theorem of arithmetic (I’ll let you people figure out how that works).
Now, if I were to create a function, f(x), such that for every natural number n, f(n) would be the set of all positive integer powers of the nth prime, it would obey all the rules I laid out in my definition: Each natural number is mapped to an infinite number set of natural numbers, and per every two numbers, the sets mapped to both numbers are disjoint. Thus, 
.
But there’s more: remember the countably and uncountably infinite sets? Well, through a simple isomorphism you could extend that to all countably infinite sets, thus rendering the whole thing even more ridiculous (and, essentially, since there is an infinite number of countably infinite sets, an infinite number of flops, each of which infinite in its own right).
If I were a Hollywood supervillian, I would have but one thing to say: “My plans are ruined! Nooooo!”