### Axiomatic Set Theory 2: The Axioms, Part One

#### Posted by Tom Leinster

*Previously: Part 1. Next: Part 3*

We’ve just finished the second week of my undergraduate Axiomatic Set Theory course, in which we’re doing Lawvere’s Elementary Theory of the Category of Sets but without mentioning categories.

This week, we covered the first six of the ten axioms: notes here.

The data to which the axioms apply is as follows:

some things called

**sets**;for each set $X$ and set $Y$, some things called

**functions from $X$ to $Y$**, with functions $f$ from $X$ to $Y$ written as $f: X \to Y$;for each set $X$, set $Y$ and set $Z$, an operation called

**composition**assigning to each $f: X \to Y$ and $g: Y \to Z$ a function $g \circ f: X \to Z$;for each set $X$, a function $id_X: X \to X$, called an

**identity function**.

The first six axioms:

Composition is associative, and the identity functions act as identities.

There is a terminal set, $\mathbf{1}$.

A function is determined by its effect on elements. That is, define an element of a set $S$ as a function $\mathbf{1} \to S$. Then whenever $f, g: X \to Y$ with $f x = g x$ for all elements $x$ of $X$, we have $f = g$.

There is a set with no elements.

For all sets $X$ and $Y$, there exists a product of $X$ and $Y$. Here “product” is defined by the usual universal property.

For all sets $X$ and $Y$, a function set from $X$ to $Y$. Here “function set” means what is usually called an exponential in category theory, and again is defined by the usual universal property.

The details of all this, together with lots of explanation, are in the notes.

## Re: Axiomatic Set Theory 2: The Axioms, Part One

I think the empty set axiom is redundant, since empty sets can be constructed from the terminal set, products, inverse images, function sets, and the subobject classifier.