Math and science::Analysis::Tao::08. Infinite sets

# 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 [...] is a function

$f : \mathcal{I} \rightarrow \bigcup_{i \in \mathcal{I}} S_i$

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 [...].

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

The Cartesian product of an indexed collection of nonempty sets is [...]:

$S_i \neq 0 \; \forall i \in \mathcal{I} \implies \prod_{i \in \mathcal{I}} S_i \neq 0$

It's worth noting that the axiom of choice is only an interesting statment when the [...].

### 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 [...].