# Axiom of Choice

There are a few different ways of presenting this axiom differing in their readability.

### 1: choice function (brilliant.org)

Let \( \mathcal{I} \) be a set (which we will use for indexing). For each \( i \in \mathcal{I} \) let \( S_i \) be a set. We call the set of these sets a collection, denoted as \( \{ S_i \}_{i \in \mathcal{I} } \). So the collection contains one set for each element in \( \mathcal{I} \).

A choice function is a function

such that \( f(i) \in S_i \text{ for all } i \in \mathbb{I} \). The axiom of choice states that for any indexed collection of nonempty sets, there exists a choice function.

### 2: Cartesian product of indexed collection (brialliant.org)

The Cartesian product of an indexed collection of nonempty sets is nonempty:

It's worth noting that the axiom of choice is only an interesting statment when the indexing set \( \mathcal{I} \) is infinite; it is easy to show that the finite case is true without using the axiom of choice.

### 3: binary predicates

Let \( X \) and \( Y \) be sets, and let \( P(x, y) \) be a binary predicate
pertaining to an object \( x \in X \) and \( y \in Y \) such that for every
\( x \in X \) there is at least one \( y \in Y \) such that \( P(x, y) \) is
*true*. [How can one assert the existance of such a predicate?] Then
there exists a function \( f : X \rightarrow Y \) such that \( P(x, f(x)) \) is
*true* for all \( x \in X \).