Arnold Zwicky's Blog

It was a morning name many days ago, but it led in so many interesting directions that I’m just now getting to post about it: Zorn’s Lemma, a remnant of my days in logic and set theory (now almost entirely forgotten).

From the lemmatist Max August Zorn, with a brush against his newspaperman grandson Eric, to Max’s wife Alice, on to the amazing musician John Zorn (no known relation to any of the above), and then to James Thurber’s The 13 Clocks.

Start with Max and a straightforward Wikipedia article:

Max August Zorn (… June 6, 1906 – March 9, 1993) was a German mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn’s lemma, a powerful tool in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, ordered sets, etc. Zorn’s lemma was first postulated by Kazimierz Kuratowski in 1922, and then independently by Zorn in 1935.

… Max Zorn was appointed as an assistant at the University of Halle. However, he did not have the opportunity to work there for long since he was forced to leave Germany in 1933 because of the Nazi policies. He emigrated to the U.S. and was appointed a Sterling Fellow at Yale University. After that, he moved to UCLA and remained until 1946. He left UCLA to become a professor at Indiana University. He held this position from 1946 until he retired in 1971.

Zorn in 1948, in the early days of his fame:

(#1)

(Obviously, Zorn’s Lemma wasn’t named by Zorn himself.)

Wikipedia on the lemma, with an explanation that probably only hard-core set-theoretical types could warm to:

Zorn’s lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:

Suppose a partially ordered set P has the property that every chain (i.e. totally ordered subset) has an upper bound in P. Then the set P contains at least one maximal element.

It is named after the mathematicians Max Zorn and Kazimierz Kuratowski.

… Zorn’s lemma is equivalent to the well-ordering theorem and the axiom of choice, in the sense that any one of them, together with the Zermelo–Fraenkel axioms of set theory, is sufficient to prove the others.

At this point, I choose to abandon Zorn’s Lemma and go with the (equivalent) Axiom of Choice, which I have in fact taught (though not for a very long time). Talking loosely, the Axiom of Choice says that if you have an assortment of non-empty sets, it is possible to choose exactly one element from each of these sets to create a “choice set”.

Digression: Whoa, you say, isn’t this obvious? Well, you’re probably thinking of starting with finite sets, and then the AC would be provable, not an assumption. But the sets you start with can be of any size, including all kinds of infinite sets.

Then there’s an issue. The AC just asserts that there is such a choice set; it doesn’t say how to find it. And indeed if you take a constructivist view of mathematics, such a set can be asserted to exist only if you have an effective scheme for constructing it — and there isn’t one; you have to add AC as a basic assumption to get on with the business of doing set theory. Which pretty much everybody does.

So much for the heavy lifting. Now, a little silliness. From Steve Carlson’s newsletter piece of fall 2009 “Max Zorn: World Renowned Mathematician and Member of the Indiana MAA Section”:

Essentially all mathematics graduate students, as well as many undergraduate math majors, at some point in their studies learn the importance of the axiom of choice and some of its equivalent versions. The author of this note [that is, Carlson, avoiding the pronoun I] was a graduate student studying topology in the 1970’s, when the current “mathematical humor’’ in my group involved riddles like the following:

Q: “What’s sour and yellow and equivalent to the axiom of choice?”

A: “Zorn’s lemon!”

Groan.

There are many affectionate reminiscences of Zorn at Indiana. And here a 1986 photo of a genial, musical Zorn:

(#2)

And from outside Indiana, there’s a sweet-sad reminiscence by Max’s grandson Eric, a Chicago Tribune writer, in a 8/18/05 column.

Alice Zorn. That brings us to Max’s wife, Alice Schlottau Zorn, who also worked at Indiana University, as an editor in anthropological linguistics. It’s been a long time, but I think Alice was the editor of the journal Anthropological Linguistics (still housed at IU) for some time.

Unfortunately, the records of these things — the history of journals, the lives of editors (especially women), and the stories of the companions of “famous” people — are poorly kept, so there’s very little I’ve been able to find about Alice in her later days. And I’ve found no photo of her.

Note: the name Zorn. Yes, it means ‘anger, rage’ in German and probably started as a nickname for a sort-tempered man. There are lots and lots of Zorns.

John Zorn. From Wikipedia:

John Zorn (born September 2, 1953) is an American avant-garde composer, arranger, producer, saxophonist and multi-instrumentalist with hundreds of album credits as performer, composer, and producer across a variety of genres including jazz, rock, hardcore, classical, surf, metal, klezmer, soundtrack, ambient and improvised music. He incorporates diverse styles in his compositions which he identifies as avant-garde or experimental.

John Zorn with his primary instrument:

(#3)

One more Zorn: Zorn of Zorna. That brings me to one of my all-time favorite books, ostensibly a children’s book, by James Thurber, with illustrations by Marc Simont: The 13 Clocks. I wrote an appreciation of the book here on 7/29/13. The briefest of summaries from Wikipedia:

The 13 Clocks is a fantasy tale written by James Thurber in 1950, while he was completing one of his other novels. It is written in a unique cadenced style, in which a mysterious prince must complete a seemingly impossible task to free a maiden from the clutches of an evil duke. It invokes many fairy tale motifs.

The hero of the tale is Prince Zorn of Zorna, who arrives in town disguised as a minstrel named Xingu.

This entry was posted on June 21, 2015 at 10:42 am and is filed under Academic life, Books, Jokes, Music, Names. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.