Intuitive idea of axiom of choice

Question

I'm currently reading a book on set theory and it gives the following formulation of the axiom of choice: Let $X$ be a non-empty set. Then there is a function $g: \mathcal{P}(X)\setminus\{\emptyset\}\rightarrow X$ such that for all non-empty subsets $M$ of $X$, $g(M)\in M$. Here $\mathcal{P}(X)$ is the powerset of $X$. My question is, what does this actually say intuitively? I've tried looking online and I can only find intuitive ideas of other formulations of AC. Thanks in advance.

Answer 1

Remember that the powerset is a collection of all subsets. We therefore have a function from subsets into the original set. The restriction $g(M)\in M$ is saying that this map is from the subset to one of its own elements. Such a function associates every subset to one of its own elements, and can be viewed as "choosing" the element from that subset. The axiom of choice is then asserting that this form of choosing can be done for any (non-empty) set $X$.

Answer 2

My way of interpreting this is as follows:

We can choose some function $f$ such that it picks an element of every subset of $A\neq\emptyset$.

This is, I believe, the most intuitive interpretation of all.

Answer 3

This is simply saying that, for any set $X$, there exists a function that picks an element from every nonempty subset of $X$.

Let's unpack it:

We have a function $g: \mathcal P(X) \setminus \{\varnothing\} \rightarrow X$. Before doing anything else, this already says "$g$ is a function that assigns an element of $X$ to every nonempty subset of $X$" (since nonempty subsets of $X$ are the domain of our function, and our codomain is elements of $X$). Since we also require that $g(M) \in M$, we're demanding that every subset is mapped to an element inside of it. So $g$ is a function that sends every nonempty subset of $X$ to an element of that set.

Answer 4

It's hard to tell you what it means "intuitively" because studying mathematics one has to break away from previous intuition, and work hard on constructing a new intuition.

But if you just write it out into words, what does a choice function on $\mathcal P(X)\setminus\{\varnothing\}$ accept? It accepts non-empty subsets of $X$; what does it return? It returns elements of $X$. And what is the property defining a choice function? $g(M)$ is an element of $M$.

So we can say that $g$ chooses an element of $M$, for every $M$ which is a non-empty subset of $X$. Hence the name, a choice function.

The axiom of choice, if so, asserts the existence of a choice function for every $\mathcal P(X)\setminus\{\varnothing\}$. Of course, if one exists, then many exist, but set theorists are modest, and they find comfort knowing there is one.

Other ways to formulate the axiom of choice which are very common are these:

1. For every family of sets $X$ such that all its members are non-empty, there exists a function $f\colon X\to\bigcup X$ such that $f(x)\in x$ for all $x\in X$.

2. For every family of pairwise disjoint non-empty sets $F$, there exists a set $T$ such that for all $A\in F$, $|T\cap A|=1$.

And of course, the axiom of choice has many many, many equivalents throughout mathematics. Some of which are very natural, others which are not as natural, and many which are quite useful to know.