# Axiom of Choice by Well ordering principle

$$\textbf{Well Ordering Principle }:$$ Every nonempty set $$X$$ can be well ordered.

$$\textbf{The Axiom of Choice}:$$ if $$\{X_{\alpha}\}_{\alpha \in A}$$ is a nonempty collection of nonempty sets, then $$\prod_{\alpha \in A} X_{\alpha}$$ is nonempty.

In the introduction of Real Analysis by Folland, it is written that: Let $$X=\cup_{\alpha \in A}X_{\alpha}$$. Pick a well ordering on $$X$$ and, for $$\alpha \in A$$, let $$f(\alpha)$$ be the minimal element of $$X_{\alpha}$$. Then $$f \in \prod_{\alpha \in A} X_{\alpha}$$.

My question is that why we need to choose the minimal element of $$X_{\alpha}$$. Since $$X_{\alpha}$$ is not empty so why we cannot say it contains an element like $$p_{\alpha}$$ and we define $$f(\alpha)=p_{\alpha}$$.

• Did you notice you didn't say what is this about? You only say what Folland says....but we don't know what for. – DonAntonio Jun 7 '19 at 20:21
• For proving the axiom of choice. – S_Alex Jun 7 '19 at 20:39
• “Why we cannot say...?” because that inference uses the axiom of choice, which is the thing we’re trying to prove. – spaceisdarkgreen Jun 7 '19 at 20:42

The axiom of choice is, essentially, the way you are allowed to move from "for every $$\alpha$$, $$X_\alpha$$ is non-empty" to the statement "choose $$p_\alpha$$ from each $$X_\alpha$$".