Zorn's lemma

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.

Background

The terms used in the statement of the lemma are defined as follows. Suppose (P,≤) is a partially ordered set. A subset T is totally ordered if for any s, t in T we have s ≤ t or t ≤ s. Any such totally ordered set T is called a chain. Such a set T has an upper bound u in P if t ≤ u for all t in T. Note that u is an element of P but need not be an element of T. An element m of P is called a maximal element (or non-dominated) if there is no element x in P for which m < x.

Note that P is not explicitly required to be non-empty. However, the empty set is a chain (trivially), hence is required to have an upper bound, thus exhibiting at least one element of P. An equivalent formulation of the lemma is therefore:

Zorns Lemma (film)

Zorns Lemma is a 1970 American structural experimental film by Hollis Frampton. Originally starting as a series of photographs, the non-narrative film is structured around a 24-letter Latin alphabet. It remains, along with Michael Snow's Wavelength and Tony Conrad's The Flicker, one of the best known examples of structural filmmaking.

Content

The opening section of Zorns Lemma is 5 minutes long. In it a woman reads an abecedary of 24 couplets from The Bay State Primer, an eighteenth century book designed to teach children the alphabet. The film is entirely black during this section.

The film's main section is silent and lasts 45 minutes, broken into 2,700 one-second units. It shows the viewer an evolving 24-part "alphabet". The section begins by presenting each letter typed on a sheet of tin foil. The alphabet is initially composed of words that appear on street signs, photographed in Manhattan. As the film continues to cycle through the alphabet, individual letters are slowly substituted with images. The first four substitutions—fire (x), waves (z), smoke (q), and reeds (y)–depict the four classical elements.