Zorn’s Lemma – An elementary proof under the Axiom of Choice

by Arjun Jain

Last year, around the same time, I had completed a summer project on Elementary Set Theory at the Indian Statistical Institute Delhi, under Prof. Ajit Iqbal Singh. The topic that took most of my time there, was the famous Zorn’s Lemma.

The statement of the Theorem is as follows:

If X is a partially ordered set such that every chain in X has an upper bound in X, then X contains a maximal element.

After my work there,  I wrote an article about this, which is posted on the arXiv.

It presents an elementary proof of Zorn’s Lemma under the Axiom of Choice, simplifying and supplying necessary details in the original proof by Paul R. Halmos in his book, Naive Set Theory. Also provided, is a preamble to Zorn’s Lemma, introducing the reader to a brief history of this important maximal principle.

Please feel free to ask me questions related to the article.