Proving Zorn's Lemma: A Guide to Understanding and Application

[quantum123]

I want to learn how to prove the Zorn's lemma.
Can anyone here help me?

 

[micromass]

The proof of Zorn's lemma requires some knowledge of set theory. Explicitly, you'll need to know about the axioms of ZFC, the replacement axiom, the axiom of choice, transfinite induction, ordinals,...

To learn about all these things, I've got two beautiful references for you:
- Introduction to set theory, by Hrbacek and Jech
- Set theory, by Jech

If you want some free materials on the internet, then I recommend staff.science.uva.nl/~vervoort/AST/ast.pdf but it's not at all an easy lecture...

 

[quantum123]

Interesting and thanks!
I have saved staff.science.uva.nl/~vervoort/AST/ast.pdf and will read it soon.
Thanks for sharing, micromass!

 

[quantum123]

LOL
Axiom 0:
i) There exists at least 1 thing, and
ii) everything is a set.

 

[Fredrik]

There's some good stuff in this thread.

 

[natmath]

You can prove Zorn's Lemma from basic set theoretic facts, without any use of transfinite induction, ordinals etc..
A good proof is given in Zorn's Lemma- An elementary proof under the Axiom of Choice http://arxiv.org/abs/1207.6698 .

 

[quantum123]

I was really stuck by Hamos' proof. Thanks for this 6 pages explanation. I will try to understand it later. BTW, I have finally understood the transfinite induction proof - in that proof you need the ordinals ..

 

[mathman]

A bit of reminiscense: when I was exposed to this material I recall that there were several statements including the Axiom of Choice and Zorn's lemma, and they were all equivalent.

 

[quantum123]

Finished reading the proof. I wonder why must the proof of such an innocent theorem be so long. From the definition of a poset, axiom of choice, one need to define initial segments, chains, towers, comparable sets, layers upon layers of abstraction to prove something like: a maximal element exist. This reminds me of the movie: Inception. You need to dream 5 levels to plant just a simple idea.