# Axiom Of Choice

It confuses people as to why the Axiom Of Choice is contentious.

Here are two posts to explain the points of view:

Roughly ...

When we have two finite sets, $A$ and $B$, can form the set of pairs:

• $\{ (a,b) : a\in A, b\in B \}$

The nice things about that is that the size of this set is the product of the sizes of $A$ and $B$, and we call this new set the product of $A$ and $B$, and we denote it as $A\times B$.

When we have three finite sets we can form $(A\times B)\times C$, or we can form $A\times (B\times C)$, or we can form the collection of triples:

• $\{ (a,b,c) : a\in A, b\in B, c\in C \}$.

These all have natural mappings between them, and again, the size of each of them is the product of the sizes of the contributing sets.

We can avoid this "duplication" issue by a simple technical trick. If we assume that all the sets are disjoint then instead of forming pairs or triples, we can form a set, taking an element from each contributor. Thus:

• $A\times B := \{ \{a,b\} : a\in A, b\in B \}$

Note that we require the sets all to be disjoint to make sure we get 2 (or $n$) element sets, and this easily generalises to any (finite) number of contributing sets.

We can clearly form the product of finitely many sets even if some are infinite, that's not a problem, and one could easily assume that it all works without any problems even if we have infinitely many sets.

And sometimes it does, but sometimes we get some results that might catch you by surprise.

• The Vitali set;

There are more, but I refer you again to these posts: