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?

  • $\begingroup$ 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
  • 1
    $\begingroup$ See the edit to my answer. $\endgroup$
    – Asaf Karagila
    Jul 6, 2015 at 13:23

1 Answer 1


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

This is also mentioned in the errata for the book.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .