Natural numbers

We now want to learn how to count. This is how it usually goes: we start from $0$ and then we keep passing to the successor, that is, $0$ is followed by $1$, which is in turn followed by $2$, which then gives its place to $3$, and so on. Since these objects are essential for mathematics, one would rightly expect them to appear in the universal theory of mathematics, i.e. Set Theory. Well, this is what they are:

$$0:=\mathrm{\varnothing }$$

$$1:=0\cup \{0\}=\{0\}$$

$$2:=1\cup \{1\}=\{0\}\cup \{1\}=\{0,1\}$$

$$3:=2\cup \{2\}=\{0,1\}\cup \{2\}=\{0,1,2\}$$

and it continues. The problem, though, is that it continues indefinitely... This means that I can't possibly just list all definitions and have you know what the natural numbers are. A more sophisticated approach is required!

The starting point is $0$, which is just empty set. Now, to get new natural numbers, you pass to the successor. The way I wrote my definitions above was to suggest what the successor operation should be. In the previous post, I explained what a property $R(x,y)$ is. If this property additionally satisfies that for all sets $x$, there exists exactly one set $y$ so that the relation $R(x,y)$ holds, then we say that $R$ is a (class-sized) operation. We can then write $R(x)=y$ instead of $R(x,y)$ and use $R(x)$ to mean "the unique set $y$ so that $R(x)=y$". Now, the successor operation that I had in mind just before this explanation is

$$S(x):=x\cup \{x\}.$$

More precisely, $S(x,y)$ holds if and only if $y=x\cup \{x\}$.

Once we've established this new notation, we can see that

$$0=\mathrm{\varnothing }$$

$$1=S(0)$$

$$2=S(1)$$

$$3=S(2)$$

and so on. We can now state the following principle: the empty set is a natural number and if some set $x$ is a natural number, then its successor $S(x)$ is also a natural number. However, this is not enough to define what natural numbers actually are! Let me elaborate on this. We'll say that a class $C$ is inductive if and only if it contains the empty set and for every set $x$ in $C$, it is necessarily true that $S(x)$ is also in $C$. Can you give me an example of an inductive class? Why $V$, of course! However, not every set in $V$ should be a natural number: for example, there should be real numbers hanging around as well! Also, we hope to somehow have the set of all natural numbers, while $V$ itself is a proper class, hence much larger.

Without further ado, from the way I described natural numbers, it should be clear that every inductive class contains all natural numbers. However, there are inductive classes which contain other stuff except natural numbers. The class of all natural numbers is itself inductive, but from what I've just said, it is contained in every other inductive class. I would like to say then that the class of all natural numbers, denoted by $\omega$, is the smallest inductive class. In other words, $\omega$ is the inductive class which is contained in every other inductive class, but there is a subtle issue here. The issue is that this formally involves saying something like: "for all classes $X$, if $X$ is inductive, then $\omega \subseteq X$". What we did here is quantify over classes, by saying "for all classes $X$". We can't really do this, because classes are objects external to the universe, which contains only sets. Since the universe contains only sets, we are allowed only to quantify over sets.

These comments are not really an issue. Considering that we want to have the set of all natural numbers and that this set is inductive, we might as well assert that there exists an inductive set. This is a new axiom, so-called Axiom of Infinity. We can now say that $\omega$ is the smallest inductive set. To say this, we formally say: $\omega$ is inductive and for every set $X$, if $X$ is inductive, then $\omega \subseteq X$". Now we have only quantification over sets. Still, we need to verify that this definition "works properly", i.e. that it gives us a set. Since $\omega$ should be contained in every inductive set, we might as well pick one, say $X_0$. We can now pick a subset $Y$ of $X_0$ by taking those $x$ from $X_0$ which have the property that they belong to every inductive set in the universe. If you recall Axiom of Comprehension, you'll realize that we can do this and that we indeed get a properly defined set in this way. We have that:

• $0$ belongs to every inductive set, so it belongs to $Y$;
• if $x$ belongs to $Y$, it belongs to every inductive set; since every inductive set is closed for $S$, we have that $S(x)$ belongs to every inductive set; this just means that $S(x)$ also belongs to $Y$;
• by definition, if $x$ belongs to $Y$, it belongs to every inductive set; this means that $Y$ is contained in every inductive set.

The purpose of listing these points was to convince you that $Y$ is in fact just another name for $\omega$, which in particular means that $\omega$ is a well-defined set! Thus we have obtained the set of natural numbers! 🙌