Filling in the picture

When you learned math as a kid, you started from the very concrete: learning to count physical objects.

The foundations of math progress the opposite way, from the vague to the specific. The first thing you need to know is the use of variables as placeholders. Partly this is because variables are powerfully expressive—eventually, you’ll need them to write really interesting statements. But the other reason is that you start out knowing nothing. The less you know, the more gaps there will be in the statements you make. Variables fill gaps.

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Now you’re ready to study propositional logic. The axioms of this field are things like,

A ∧ B ⟹ A

“If A and B, then A.”

The variables are placeholders for statements. The purpose of an inane-sounding axiom like this one is to tell precisely what “and” means in logic.

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Next is the predicate calculus. Logic had variables that stood for statements; predicate calculus offers several kinds of statements you can write and plug into those logic variables. The variables of predicate calculus stand for two kinds of thing:

• objects, the actual things we will eventually get around to talking about, like numbers, shapes, points, functions, and so on; and

• predicates, which are statements about objects: “x is even”, “s is convex”, “p has integer coordinates”, and so on.

But predicate calculus does not by itself say what the objects are. You can take predicate calculus and apply it in a universe where there the objects are points, or where the objects are functions, and it will work equally well in either setting.

An axiom of the predicate calculus:

(∀x . P(x)) ⟹ P(t)

“If for all x, P(x), then P(t).”

This axiom is telling you what “for all” means. x stands for an object; P is a predicate. Note that we are still using the language of logic, but as we proceed, the variables are getting rather more specific, and we’re adding new vocabulary.

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Next is set theory. This is a particular kind of universe for you to use predicate calculus with. Whatever we’re talking about in math, we will want to talk about sets of those things: sets of points, sets of numbers. It turns out you’ll also very soon want to talk about sets of sets of things, so set theory lets you build sets of sets, sets of sets of sets, and so on without end. The variables in set theory stand for sets.

An axiom of set theory is:

∀S . ∃T . ∀x . x ∈ T ⟺ x ∈ S ∧ P(x)

“Given a set S and a predicate P, there exists a set that contains exactly the elements of S that satisfy P.”

The surprising thing is that now you’re done.

The universe of sets contains sets that have exactly the same structure as: the integers; the Cartesian plane; the functions on real numbers; and so on. It contains basically everything mathematicians study*, so there's no need to go on adding axioms**. The picture is*** filled in.

*not totally true

**except when there is, or when mathematicians just feel like it

***never