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I am trying to prove the following:

Every nonempty $X \subset \mathbb N$ has an $\in$-minimal element.

Proof: Take some $n \in X$. Then $\min \{ n \cap X \} = \min \{ X \}$.

Is this ok or should it be more detailed?

Frunobulax
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A student
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  • Hmm. I would suggest approaching it by contradiction. Assume that it doesn't have a least element. Then this implies that for any $x\in X$ there is an $\varepsilon >0$ such that $x-\varepsilon$ is in X. Keep going – Eoin Sep 12 '14 at 06:08
  • ok I will try that. The reason I took that direction is that this is the what Jech recommanded in his book for this exercise (1.7 chapter 1) – A student Sep 12 '14 at 06:22
  • Another thing. It is not clear to me why do I immidiately get "-1" score on this question. I am new in this stuff. giving a serious effort for the proof an exercises. and also, most of them even turned out to be ok so I am improving as time goes on. Try to give some credit here. Think of the very first time in your life you saw these definitions – A student Sep 12 '14 at 06:22
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    It should be more detailed. You are hinting at induction and the statement you're trying to prove, called the 'well-ordering principle', is actually equivalent to the principle of mathematical induction, so you should probably emphasize the role of induction in your solution. – Zavosh Sep 12 '14 at 06:23
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    I assume the downvote is because your question is extremely brief and not clear. You don't mention what logical system you're working in, and the only way to tell you are talking about a set-theoretic answer is to look at the tag. – Zavosh Sep 12 '14 at 06:25
  • @Student_Afeka I didn't downvote you. But people on this forum tend to downvote more often when less information is given. – Eoin Sep 12 '14 at 06:28
  • I understand now. Thank you for your clarifications. – A student Sep 12 '14 at 06:36
  • How do you know that $n \cap X$ has a $\in$-minimal element (even if it is nonempty)? – user642796 Sep 12 '14 at 06:46
  • I know that $n$ has a minimal element since it is a finite set. So, $X \cap n$ is finite and has a minimal element. The difficult thing here is to understand what am I allowed to assume and what I am not.. Any enlightenment regarding that? Thank you – A student Sep 12 '14 at 06:52
  • I suggest you see: http://math.stackexchange.com/questions/358979/proving-the-so-called-well-ordering-principle – Rustyn Sep 12 '14 at 06:59
  • I see now.. will have a look – A student Sep 12 '14 at 07:01
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    One reason for the downvote (wasn't from me) might be that @Student_Afeka has asked half a dozen similar questions (which all look like elementary exercises from the first chapters of Jech's book) in a few hours. – Frunobulax Sep 12 '14 at 07:16
  • They are similar but far from being the same. And I thought other students could learn from it – A student Sep 12 '14 at 07:18

1 Answers1

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As to whether your answer is OK in the sense of being correct, I'd say yes.

As to whether it's detailed enough, I'd say: That depends. There are a lot of seemingly "easy" questions like these in the beginning of set theory and the most important point is not to find some kind of "acceptable" answer but rather to watch out for what you already know (by axioms or things you've already proved) as opposed to what you only think you know. I'd therefore recommend that in the beginning you are as detailed as possible justifying each step in your proof. And for others to help you you need to tell them exactly this: What you already know and are therefore allowed to use (like from previous exercises). Once you get the hang of it, you can get a bit more loose.

Here for example for $\min\{n \cap X\}$ to make sense you already need to know that $x \in \mathbb N$ implies $x \subseteq \mathbb N$. And then you need to know that $n$ (viewed as a set of natural numbers) has an $\in$-minimal element. For someone trained in set theory, these are all obvious facts, but they won't know in which order you've been taught such things, i.e. whether you are allowed to use these facts or not. That makes it difficult to help you.

Frunobulax
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  • I see. Considering these are the first exercises of Chapter 1 of Jech's book. Am I allowed to assume the ZF axioms in an exercise like that? Am I allowed to assume anything else? Because it is not detailed in the exercise what am I allowed to assume – A student Sep 12 '14 at 07:36
  • If I recall correctly, Jech doesn't present all axioms at once (e.g. the axiom of foundation/regularity comes relatively late). So, for an exercise you usually should be allowed to use A) all axioms introduced so far, B) all theorems and lemmas proved so far, i.e. up to the exercise, and C) previous exercises. All this of course unless the specific exercise says something different like "prove this without using the foo axiom". – Frunobulax Sep 12 '14 at 07:41
  • I see now. In this case, I guess I should have not use knowledge about ordinals. So, assuming I am allowed to use the axiom of infinity, the right proof is probably the one Peter L. Clark gives in here: http://math.stackexchange.com/questions/358979/proving-the-so-called-well-ordering-principle. Thank you! – A student Sep 12 '14 at 08:41