The Axiom of Infinity
In my previous post introducing the concept of Set Theory, we discussed one method for constructing the Natural numbers– a method often referred to as a Von Neumann construction. Using that method, we start with the Empty Set () and then systematically build the Natural numbers by following a rule. As described in that post, this was a step-wise process: look at a number, find its successor, look at the new number, find its successor, repeat ad infinitum. Now, obviously, given a finite amount of time there would be no way to perform this process enough times to generate every Natural number, since every new number we create would still have yet another number succeeding it.
But what if we want to discuss the whole set of Natural numbers?
As we just noted, we cannot construct the Natural numbers in a step-wise manner in order to get all of them. However, mathematicians like Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem devised a very clever way to take the very same ideas from our step-wise construction in order to discuss a whole, completed set. We refer to this notion as the Axiom of Infinity, and it is one of the premises which underlies the vast majority of modern mathematics.
Written in formal language, the Axiom of Infinity can be stated thusly:
If that looks completely unintelligible to you, do not fret! The meaning behind those symbols is not difficult to understand. Let’s look at each section, in turn.
First, let’s ignore everything inside the parentheses and just look at the major statement of the axiom: . In plain English, this simply means, “There exists at least one set which we will refer to as I which has the following properties.” Since this is an axiom, we are saying that we simply presume this statement to be true. We are not wondering whether it is true or trying to prove that it is true. We simply presume that it is true and then follow the logical implications of that presumption.
Now, let’s add in the next phrase of our axiom: . This is read as, “There exists some set, I, such that the Empty Set is an element of I.” A set, as we’ll recall, is simply a collection of elements. This phrase of the Axiom of Infinity just tells us that at least one of the elements of our set, I, is the Empty Set. So, now we presume that is in I— again, we do not find or prove or investigate this statement; we simply presume it.
The next symbol in the axiom, , just means, “and.” If I were to write the formal statement , this would be read as, “A and B.” Thus, the statement means, “There exists some set, I, such that the Empty Set is an element of I and such that B is true.”
Now we will look at , which means “for all x in I the following is true.” This tells us that we are about to learn some property which is shared by every single element in I. The phrase inside those parentheses will detail that property.
Zooming in on that final phrase, we have . This is read as, “the set which is the union of x and the set containing only x as an element is an element of I.” So what does this mean? Believe it or not, it is actually a concept with which we are already familiar. Recall from the previous post on Set Theory that numbers can be thought of as sets, and that the way in which we find the Successor to a number– that is, the next number in the sequence– is to create a new set which has all the elements which are in our number, but also contains that number as an element. A union is a set which contains all of the elements of two (or more) other sets but does not contain any other elements. So, the phrase is just another way of saying, “the Successor to x.”
Putting this together with our previous statement, we now see which means, “for every element, x, which is in I, the Successor to x is also in I.” Notice the big difference between this statement and what we did in the previous post on Set Theory. Here, we are not playing a step-wise game, constructing new numbers at each stage. Rather, we are saying that all of these Successors are already elements of I. They don’t need to be added to the set– they already exist in the set.
So, putting everything together, we now see that means, “There exists at least one set, I, such that the Empty Set is an element of I and such that, for every element of I, the Successor to that element is also an element of I.” Now, since we know that every element has a Successor which is also an element, we can quickly deduce that there must be an infinite number of elements contained within I. So now, we get to the real heart of the Axiom of Infinity. If we were to paraphrase the implications of this axiom using common English, rather than using this very formal language, we get a strikingly simple statement:
There exists at least one set with an infinite number of elements.
This Axiom of Infinity is one of the two most controversial premises which underlie mathematics (the other being the Axiom of Choice, which I will save for another day). Though it is not nearly so hotly debated now as it was in the early 20th Century, there remains a small contingent of philosophers of mathematics who take great exception to the Axiom of Infinity. The reason for this is that, as I stressed earlier, axioms are not statements which are proven or investigated; rather, they are statements which are simply presumed to be true. Because of this fact, it is generally preferred that a system’s axioms should be things which are almost universally agreed upon– things which are self-evident or which are extremely difficult to deny. There are some who quite vehemently claim that this is not at all the case for the existence of an infinite set. The mathematician L.E.J. Brouwer famously opposed the Axiom of Infinity and had heated arguments with other eminent powerhouses in the field like David Hilbert and Abraham Fraenkel. One of the modern mathematicians who finds the Axiom of Infinity to be problematic is Norman Wildberger, whose claims have been a subject of discussion on this blog, before.
I don’t find their arguments very compelling. I have a great deal of respect for these mathematicians, and I will fully admit that they have accomplished more for mathematics than I ever will; but I find the Axiom of Infinity to be a beautiful and fascinating thing. Such a small statement opens up entire universes of mathematics which could not be explored without it. It elegantly lays a foundation for both the work which preceded its invention and that which succeeded its creation. It has fueled the field of mathematics for nearly one hundred years and is likely to continue in that way for the vast, foreseeable future. So, let us continue to presume that there exists at least one infinite set. Let us continue to explore the Axiom of Infinity.