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 \( x \in X \).
Example
Another way of viewing AC was considered here: https://twitter.com/gro_tsen/status/1471204761747738628?s=21
