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How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous proof. It would be nice if someone could direct me to a book containing such a proof.

EDIT: I am looking for literature where this is proved STRICTLY from the ZF axioms.

Najib Idrissi
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user159517
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1 Answers1

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The proof is simple enough that you don't need a book. You just prove it by induction. The base case (one set) is trivial, and the inductive case is not much harder, using the fact that $$\prod_{i = 1}^{n+1} X_i \cong \left (\prod_{i=1}^n X_i\right ) \times X_{n+1}.$$

Carl Mummert
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  • It is simple because it is not rigorous, meaning that it's not a direct formal derivation only using the axioms, which is what I'm looking for, more precisely, a book containing such a derivation. – user159517 Jan 25 '15 at 15:42
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    @user159517: I looked through several books but found nothing. This is the sort of thing that is left as a remark "Prove by induction" or an exercise to the reader. Do you "accept" a proof using the induction theorem (on natural numbers) as a valid proof? Because based on the induction theorem it is an easy proof, and formal proofs of induction (as a specific principle, or an application of transfinite induction limited to $\omega$) is easier to find references to. – Asaf Karagila Jan 25 '15 at 17:01
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    Induction in ZF is particularly easy: $\omega$ is defined as the intersection of all sets of ordinals that contain $0$ and are closed under successor. So we just have the prove that the set of ordinals $\alpha$ that are zero or for which products of length $\alpha$ are nonemmpty, is closed under successor. – Carl Mummert Jan 25 '15 at 17:23
  • @AsafKaragila thank you for your efforts. I wanted to use this fact in a proof that Tychonoff's theorem implies the Axiom of Choice. Unfortunately, I was unable to find any reference containing a strictly formalized proof, and in my eyes, any other kind of reference would miss the point. But if you could direct me to a book that is written in this highly formalized style and contains this question as an exercise I think that would suffice, I might have to go through all of it and do the proof myself, but I am sure it would help. – user159517 Jan 26 '15 at 00:47
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    @user159517: Halbeisen's book is very formal; as is Kunen's book (at least the old edition). I'd start with the latter. – Asaf Karagila Jan 26 '15 at 00:56