Confusion about Zorn’s lemma.

elementary-set-theoryorder-theory

I'm reading the book Notes on Lattice Theory, J. B. Nation and am confused about the proof of two equivalent axioms of AC (in Appendix 2: The Axiom of Choice, p137), the two axioms are:

  • Zorn's Lemma

    If every chain in an ordered set $P$ has an upper bound in $P$, then $P$ contains an maximal element.

  • Hausdorff Maximality Principle

    Every chain in an ordered set $P$ can be embedded in a maximal chain.

I have only basic knowledge about set theory and get confused about the proof from Zorn's Lemma to Hausdorff Maximality Principle:

Given an ordered set $P$, let $Q$ be the set of all chains in $P$, ordered by set containment. If $\{C_{\alpha}: \alpha \in A\}$ is a chain in $Q$, then $\bigcup C_{\alpha}$ is a chain in $P$ that is the least upper bound of $\{C_{\alpha}: \alpha \in A\}$. Thus $Q$ satisfies the hypothesis of Zorn's Lemma, and hence it contains a maximal element $C$, which is a maximal chain in $P$.

Isn't it possible that $\{C_{\alpha}: \alpha \in A\}$ is an infinite set? If possible, is there any constraint on the use of a union of infinite many sets? Like my proof below:

  • For any chain $C$ of $\mathbb{N}$, the sum of every element of $C$ is an upper bound of $C$, can I use Zorn's Lemma here and say $\mathbb{N}$ has a maximal element?

And I always feel lost when proofs have something to do with infinite stuff, is there any book about infinity? Many thanks!

Best Answer

In principle, indeed, your sum as an upper bound is very similar to the union as an upper bound, but as pointed out in the comment, the union of infinitely many sets is always a set, but the sum of infinitely many natural numbers may not be a natural number.

However, it's worth noting that taking the union of sets also has its limit. For example, we cannot take the union of all sets to get a final set without running into Russell's paradox. In ZF, we can form the set $\cup_{i\in I} S_i$ when the index $I$ is itself a set. In this particular case, we don't have trouble in forming the union of $\cup_{\alpha\in A}C_{\alpha}$ because $A$ is a set. Roughly $Q$ can be separated from the power set of $P$, hence not large enough to be a proper class.

In other words, just like we have to define integer additions, and cannot extend the operation to infinitely many integers, we also need to define set union and carefully study when this is allowed.

Related Question