Zorn's Lemma Implies Zermelo's Well-Ordering Theorem/Mistake

Source Work

Prologue

$2$. Orderings: $(\text P.3)$

Every nonempty set $X$ can be well-ordered.

Consider the collection $\WW$ of well orderings of subsets of $X$.

Such well orderings may be regarded as subsets of $X \times X$, so $\WW$ is partially ordered by inclusion.

It is easy to verify that the hypotheses of Zorn's lemma are satisfied, so $\WW$ has a maximal element.

It is insufficient for the well-orderings to be ordered by inclusion $\subseteq$.

We additionally need some kind of stability of the initial segment through a chain, for example as established in the main proof.

Proceeding with just $\subseteq$ as partial ordering of $\WW$, consider the sequence of well-ordered sets:

$W_n = \set {-i: 0 < i \le n} \subseteq \Z$

endowed with the standard ordering.

Then $\sequence {W_n}_{n \mathop \in \N}$ is a chain in $\WW$, but there is no upper bound, for:

$\ds \bigcup_{n \mathop \in \N} W_n = \set {-i: i \in \N_{>0} } = \Z_{<0}$

has no smallest element with respect to $<$.

Hence the hypotheses of Zorn's Lemma are not satisfied.