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I was reading the proof of the fact that every nonzero proper ideal in a dedekink domain factors uniquely into a product of prime ideals. I was stumped by the beautiful application of Zorns Lemma to this prove theorem. It didn't even occur to me slightest that Zorn's Lemma can be used to prove a result of this form.

Now my question is what are the signs one should look for in a statement which may suggest that Zorn's Lemma might be useful to prove that statement? In some cases you can easily see the use of zorn's lemma for example when using it to the prove that a every ring has a maximal ideal. I'm more interested in the non-obvious cases like the one mentioned above.

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My experience with axiom of choice equivalents is typically that their use tends to appear naturally.

For example, if I wanted to prove every vector space has a basis, I might get the idea that I can "construct" a basis by taking a linearly independent set and enlarging it until I can't do so anymore. Once I have that idea, appealing to Zorn's lemma to show such a construction actually works is a very natural thing to do.

Admittedly, I have much more experience with transfinite iteration, which is somewhat more obvious when it's needed. To prove the same theorem, I think in terms of starting with the empty set and repeating the procedure "Choose an element not in the span of my set. Add it to the set." until it can't be done anymore. The need for transfinite iteration is obvious since a vector space may be infinite dimensional and I need to do this infinitely many times.

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+1: As an analyst, I find myself using transfinite induction on many occasions. Every time I have to use it, I try to find the corresponding Zorn’s Lemma argument. – Haskell Curry Feb 3 '13 at 6:51

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