An axiom is traditionally defined as a self-evident truth. However, a more accurate modern description is that an axiom is a statement that is accepted as true, without proof, within some logical system, and used as a starting point to allow other statements to be proved within the system. The most famous examples of axioms are those of Euclid, but the more relevant ones to modern mathematician are the Zermelo-Fraenkel (ZF) axioms of set theory. The ZF axioms are powerful enough to prove the vast majority of theorems that mathematicians are interested in.
The word “set” has a more particular meaning in mathematics than when it is used colloquially. The best way to think about a mathematical set is as a question which can be asked about any object. For each object, the answer will always be either “yes”, in which case the object is in the set, or “no”, in which case the object is not in the set. Normally you would think of a set as a physical collection of objects, such as coins in your wallet, but this can be misleading. First of all, it might not occur to you, if you think of sets this way, that there is absolutely no limit on their size. Secondly, it is meaningless to ask how many instances a set contains of a particular object, even though your wallet might have multiple instances of a particular kind of coin in it. The question has to be answered as either “yes” or “no”. Objects are either in the set, or not in the set. Finally, there is no limit in what a set can contain. Anything you can ask a question about can be in a set. Sets can contain other sets. Sets can even contain themselves!
This precise concept of a set is captured by the first ZF axiom, which is called the axiom of extensionality. It says that for all intents and purposes, two sets are exactly the same object if and only if each of them contains every object in the other one. In formal logical language, given two sets and ,
The ancient Greeks would probably have been satisfied with this axiom; it’s self-evident in that it’s essentially a statement of what the word “set” means.
Early set theorists completed their set theory by adding an axiom schema, called the axiom schema of comprehension. Before I explain what an axiom schema is, I’ll tell you what this axiom says: it says that for every conceivable logical formula, there is a set whose members are precisely the objects that satisfy the formula. Basically, for every collection of objects you might want to work with, there’s a set containing just those objects. The reason why this isn’t a true axiom is that an axiom is supposed to be a logical formula itself, and you can’t talk about logical formulae within logical formulae! Well, actually, there are systems of logic where you can do this, but it gets really complicated and it would certainly be better if we didn’t have to resort to such advanced systems of logic. Thankfully, there’s an easy way to get around the problem. We just say that for every logical formula, there’s an individual axiom which says that there’s a set whose members are the objects that satisfy the formula. This means our theory has an infinite number of axioms, which Euclid probably wouldn’t be too happy with. This doesn’t faze modern mathematicians too much though; I guess they leave it up to the philosophers of mathematics to argue over whether infinity is real.
[There is, actually, a way to finitely axiomatise NBG set theory, which is essentially the same as ZF but slightly stronger. Look it up if you’re interested.]
Note that we haven’t actually including any other axioms stating that objects exist. If we only have these two axioms, the only objects in our logical universe are, in fact, sets created by the axiom schema of comprehension. You might wonder, then, how there can be any objects at all, since sets have to contain objects themselves, don’t they? The axiom schema of comprehension doesn’t prove the existence of any sets which contain themselves, because there isn’t a logical formula that talks about the set if it doesn’t actually exist. The possibility that a set containing itself could exist isn’t ruled out, we just can’t prove that they exist. However, the axiom schema of comprehension does prove the existence of a set containing no members, which is called the empty set and denoted . To see this, just let the logical formula be any one which involves a contradiction and is therefore false applied to every object. Once the empty set’s existence is proved, we can prove the existence of an infinite number of more complex sets, such as the set which contains only the empty set, the set which contains both of the afore-mentioned sets, and so on.
It would be very nice if all set theory required was these two axioms. Unfortunately, Bertrand Russell came along in 1901 and ruined everyone’s fun by showing that the axiom of comprehension proves a contradiction. More on that tomorrow.