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Topic: Why don't texts give a stronger version of Zorn's lemma?
Replies: 6   Last Post: Nov 20, 2013 12:09 PM

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Registered: 7/12/10
Re: Why don't texts give a stronger version of Zorn's lemma?
Posted: Nov 20, 2013 3:34 AM
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On Wednesday, November 20, 2013 12:52:40 AM UTC, quasi wrote:
> Paul wrote:

> >
> >Why isn't the standard version of Zorn's lemma as follows? :
> >Let X be a partially ordered set such that every well-ordered
> >subset has an upper bound. Then X has a maximal element.
> Because in practice, in the given context, the fact that a given
> chain has an upper bound typically depends only on the fact that
> it's a chain, not on whether it's well-ordered.

This is an excellent reply. Whenever I've seen Zorn's lemma used, the weaker version has been sufficient.

I only found out about the stronger version fairly recently, and I found it interesting that the weaker version was more standard. There are other cases in maths that I know of where theorems have two versions, and where the weaker version is more standard (more often quoted and more often used) but in these other cases, the stronger version is significantly harder to prove. Here we have a case where the stronger version (assuming AC) isn't much harder to prove.

[Example of weak and strong theorems where the weak result is more standard: Most references to the complex analysis theorem that the Cauchy Riemann equations imply differentiability assume that the partial derivatives are continuous -- in other words, Looman-Menchoff isn't used all that frequently.]

Paul Epstein

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