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I have an exercise in a book that asserts that if a set $S$ has a choice function on it, then so does the union of all its elements $\bigcup S$ (without assuming the axiom of choice). I, however, have serious doubts that this statement holds and I even have an argument against it. However, since I am up against the word of a textbook, I cannot be sure. I may just be missing something obvious. Hence, I want to verify my hunch.

First a little clarification of my terminology:

  • By a choice function $C$ on a set $S$, I mean a function $C : \wp(S)\setminus\{\emptyset\} \to S$ such that $C(\sigma) \in \sigma$.
  • For a set $S$, $\bigcup S = \{x \in \mathcal{U} : \exists y \in S\text{ s.t. } x \in y\}$ ($\mathcal{U}$ is the universal class) i.e. it is union of all the elements of $S$.

Now, my argument against the book's assertion is this: if the axiom of choice (AOC) were false, then there would be a set $\Sigma$ which has no choice functions on it. Consider, now, the singleton set $\{\Sigma\}$; it clearly has a choice function on it but its union doesn't as $\bigcup \{\Sigma\} = \Sigma$. Hence, if AOC were false, then I can prove this statement false using the other axioms of set theory. Thus, contrapositively, if I could prove this exercise true using just the other axioms of set theory, I could prove AOC too which seems ridiculous! Is my reasoning correct? Or am I mistaken somewhere?

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  • $\begingroup$ What book is this from? $\endgroup$
    – Asaf Karagila
    Jul 6, 2015 at 13:12
  • $\begingroup$ Smullyan and Fitting's Set Theory and the Continuum Problem. $\endgroup$
    – balddraz
    Jul 6, 2015 at 13:15
  • $\begingroup$ Thanks for the confirmation. I will cross out that exercise in the book. For everyone's reference, it is exercise 4.3 of Chapter 4. $\endgroup$
    – balddraz
    Jul 6, 2015 at 13:19
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    $\begingroup$ See the edit to my answer. $\endgroup$
    – Asaf Karagila
    Jul 6, 2015 at 13:23

1 Answer 1

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No. Of course not. Your proof is perfectly valid.

This is also mentioned in the errata for the book.

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