Original proof of Zorn's Lemma

Question

On the wiki page for Zorn's Lemma it says that this lemma was

Proved by Kuratowski in 1922 and independently by Zorn in 1935

but then it says:

Zorn's lemma is equivalent to the well-ordering theorem and the axiom of choice

This seems to be saying that Kuratowski and Zorn proved the axiom of choice.

That doesn't sound right. I guess it means that Kuratowski and Zorn presented a proof of what we now call Zorn's Lemma which uses the axiom of choice as an axiom.

Did they also prove the converse, i.e. that the lemma being true implies that the axiom of choice must be true?

Answer 1

The question can be answered browsing:

• Gregory H. Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence (1982, also Dover reprint.

[page 167] the concept of maximality, though not the name, emerged as a distinct notion only with Hausdorff's researches of 1907 on ordered sets of real functions. In particular, he applied the Well-Ordering Theorem to establish the existence of a pantachie, i.e., a maximal set of real-valued sequences ordered according to which sequence was eventually greater.

When Hausdorff gave a second proof for this result two years later, he based it on a new set-theoretic theorem which he deduced from the Well-Ordering Theorem.

Hausdorff's theorem did not receive the attention that it deserved. When he returned to it in his textbook on set theory [1914], he phrased it in two variant forms, each of which was later termed Hausdorff's Maximal Principle. However, it is important to realize that, despite the name later given to them, he did not propose either of these variants as an axiom or a principle:

(3.4.2) Every partially ordered set $M$ has an ordered subset $A$ that is greatest [$\subseteq$-maximal] among ordered subsets of $M$.

(3.4.3) Every partially ordered set $M$ has an ordered subset $A$ that is greatest [$\subseteq$-maximal] among ordered subsets and also includes $B$, a given ordered subset of $M$. [1914, pages 140-141]

Hausdorff, who deduced both variants from the Axiom of Choice, made no attempt to show their equivalence to it. Nor did he apply them to obtain further results. Like their predecessor, these two theorems attracted little attention from Hausdorff's contemporaries. As a consequence, maximal principles were rediscovered independently several times, the most important instances occurring in articles by Kazimierz Kuratowski [1922] and Max Zorn [1935].

[page 220] When the second edition of Hausdorff's Grundzuge appeared in 1927, he considered such formulations once again. There he introduced for the first time the term "maximal set" (Maximalmenge) to describe what is here called an $\subseteq$-maximal element, i.e., a member $B$ of a family of sets such that $B$ is not properly included in any member of the family. On that occasion he stated a maximal principle, and its corresponding minimal principle, by means of transfinite ordinals [...]. Thus his maximal and minimal principles took the following form:

(4.4.1) If $A$ is a non-empty family of sets and is extendable above by limits, then $A$ contains a Maximalmenge [$\subseteq$-maximal element] $B$.

(4.4.2) If $A$ is a non-empty family of sets and is extendable below by limits, then $A$ contains a Minimalmenge [$\subseteq$-minimal element] $B$. [1927, page 174]

Although the young Warsaw mathematician Kazimierz Kuratowski, had established such a minimal principle five years earlier, in 1927 Hausdorff remained unaware of Kuratowski's principle. Finally, Hausdorff mentioned that (4.4.1) could be used to obtain a proposition deduced earlier in his book from the Well-Ordering Theorem: In every metric space $X$ and for every positive real $r$, there is an $\subseteq$-maximal subset $Y$ of $X$ such that the distance between any two points in $Y$ is at least $r$. In all probability he did not suspect that (4.4.1) or (4.4.2) was equivalent to the Axiom [of Choice].

Hausdorff believed that these two theorems could be demonstrated only by means of transfinite ordinals [1927, 173], [...] however, Hausdorff was mistaken, since Kuratowski had demonstrated one such principle, by using the Axiom but no ordinals, in the style of Zermelo's second proof of the Well-Ordering Theorem.

[page 223] topologist R.L. Moore, [in 1932] published a volume on the foundations of point-set topology, and based his treatment on postulates containing the undefined terms "point" and "region." At the beginning of his book he discussed the Axiom of Choice, stated as Zermelo had done in 1908, which Moore included "among the fundamental propositions of the logic of classes". Much later in his book, he established a theorem which was in effect a minimal principle [1932,84]:

(4.4.5) Suppose $G$ is a [non-empty] family of sets. If, for every subcollection $H$ of $G$ ordered by inclusion, there is some $A$ in $G$ such that $A \subseteq B$ for every $B$ in $H$, then some member $M$ of $G$ has no proper subset in $G$ [i.e., $M$ is $\subseteq$-minimal].

However, he did not go on to obtain any consequences of (4.4.5), which he had deduced from the Well-Ordering Theorem.

Finally, Zorn's Lemma, the maximal principle which has become the most widely known and about whose origins we are the best informed, was published in 1935:

(4.4.6) If $A$ is a family of sets such that the union of every chain $B \subseteq AB$ is in $A$, then there is a member $C$ of $A$ which is not a proper subset of any member of $A$ [i.e., $C$ is $\subseteq$-maximal].

In contrast to the mathematicians discussed thus far, Max Zorn did not regard (4.4.6) as a theorem but as an axiom. He named it the "maximum principle" and earnestly hoped that it would supersede the Well-Ordering Theorem in abstract algebra. [...] Evidently Zorn was not acquainted with previous maximal principles. Many years later he acknowledged having read [Kuratowski 1922] but claimed not to have noticed any maximal principle there. Nevertheless, we may wonder whether Kuratowski influenced Zorn (as Hausdorff might well have influenced Bochner) in a subliminal way.

What were the beginnings of Zorn's principle? According to his later reminiscences, he first formulated it at Hamburg in 1933, where Claude Chevalley and Emil Artin then took it up as well. Indeed, when Zorn applied it to obtain representatives from certain equivalence classes on a group, Artin recognized that Zorn's principle yields the Axiom of Choice. By late in 1934, Zorn's principle had found users in the United States who dubbed it Zorn's Lemma. In October, when Zorn lectured on his principle to the American Mathematical Society in New York, Solomon Lefschetz recommmended that Zorn publish his result. The paper appeared the following year.

Answer 2

It means that if you assume the truth of the Axiom of Choice then you can prove the well-ordering theorem, and if you assume the well-ordering theorem then you can prove Zorn's Lemma and if you assume Zorn's Lemma you can prove the Axiom of Choice. Take your pick. Each is equivalent to the other, but the only thing proved is their equivilence.