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I know there are several similar posts that may be asking the same thing, but I wasn't able to find anything that could help from the previous posts.

I need to show the following statements are equivalent:

  1. The product of a family of nonempty sets indexed by a nonempty set is nonempty.
  2. Every set $S$ has a choice function.

Where a choice function of $S$ is defined to be a function $f$ from the set of all nonempty subsets of $S$ to $S$ such that $f(A)\in A$ for all $A\neq \emptyset, A\subset S$.

I think I have $(2) \implies (1)$:

Suppose every set $S$ has a choice function. Then $S$ has a function $f$ from the set of all nonempty subsets of itself to itself such that $f(A)\in A$ for all $A\neq \emptyset, A\subset S$. Let $S$ be the product of a family of nonempty sets indexed by a nonempty set $I$. Then by assumption, $S$ has a choice function such that $$f(\Pi_{i\in I}A_i)\in \Pi_{i\in I}A_i,$$ and therefore, the product of a family of nonempty sets indexed by a nonempty set is nonempty.

I'm less sure of $(1)\implies (2)$. Any help would be greatly appreciated.

TheMobiusLoops
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  • I'm not sure, but we know that there will be a least ordinal in which the set will live (in our power set model of ZFC), perhaps we can take the product of this ordinal and our target set, and since the result will be non-empty and should be orderable (due to the properties of ordinals), we can pick the least. – ZirconCode Sep 05 '16 at 21:41
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    Like http://math.stackexchange.com/questions/101004/the-axiom-of-choice-and-the-cartesian-product or http://math.stackexchange.com/questions/625277/axiom-of-choice-and-cartesian-products or one of other questions on the site? – Asaf Karagila Sep 05 '16 at 21:41
  • The statements seem equivalent from the start. An element of a product is a function from the index set to the product (and you call that function a choice function). – H. H. Rugh Sep 05 '16 at 21:43

1 Answers1

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In order to prove that $(2)$ implies $(1)$, you must start with a product $\prod_{i\in I}A_i$ of non-empty sets $A_i$ indexed by a non-empty set $I$. Your task is then to find a set $S$ such that a choice function for $S$ somehow produces an element of $\prod_{i\in I}A_i$. The natural set to try is $S=\bigcup_{i\in I}A_i$, since it has every $A_i$ as a subset. Now let $f$ be a choice function for this $S$, so that $f(A_i)\in A_i$ for each $i\in I$. We can then use $f$ to define

$$g:I\to S:i\mapsto f(A_i)\;,$$

and I leave it to you to verify that $g\in\prod_{i\in I}A_i$.

It’s for the other implication that you want to start with an arbitrary non-empty set $S$ and somehow use $(1)$ to get a choice function for $S$. I suggest that you apply $(1)$ to family

$$\big\{A_X:X\in\wp(S)\setminus\{\varnothing\}\big\}$$

to conclude that

$$\prod_{X\in\wp(S)\setminus\{\varnothing\}}A_X\ne\varnothing\;;$$

from there it’s a very short step to the desired choice function.

Brian M. Scott
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