What are zigzag theories, and why are they called that? I've encountered the term zigzag theory while randomly clicking my way through the internet. It is given here. I haven't been able to find a clear explanation of what constitutes a zigzag theory. Here, it is said that they have to do with non-Cantorian sets, which, as I understand, are sets that fail to satisfy Cantor's theorem. The article also says that New Foundations is a zigzag theory, but I don't see that it says why exactly that is so. I have gone through the Wikipedia article on New Foundations, and there's nothing about zigzags in it.
So what makes a zigzag theory? And what's zigzagging about it?
 A: The following is taken from (it is probably important to note the year of the article):

B. Russell, On some difficulties in the theory of transfinite numbers and order types, Proc. London Math. Soc. Ser.II, Vol.4 (1907), No.1, pp.29-53, journal link.

We first take a definition from the article.

A propositional function of $x$ is any expression $\phi ! x$ whose value, for every
  value of $x$, is a proposition; such is "$x$ is a man" or "$\sin x = 1$."

Now onto one of the theories discussed in the paper.

In the zigzag theory, we
  start from the suggestion that propositional functions determine classes
  when they are fairly simple, and only fail to do so when they are complicated
  and recondite.  If this is the case, it cannot be bigness that makes
  a class go wrong; for such propositional functions as "$x$ is not a man"
  have an exemplary simplicity, and are yet satisfied by all but a finite
  number of entites.  In this theory, as well as in the theory of limitation
  of size, we define a predicative propositional function as one which determines
  a class (or a relation, if it contains two variables); thus in the
  zigzag theory the negation of a predicative function is always predicative.
  In other words, given any class $u$, all the terms which are not members of
  $u$ form a class which may be called the class not-$u$.
If now $\phi ! x$ is a non-predicative function, it follows that, given any
  class $u$, there must either be members of $u$ for which $\phi ! x$ is false, or
  members of not-$u$ for which $\phi ! x$ is true.  (For, if not, $\phi ! x$ would be true
  when, and only when, $x$ is a member of $u$; so that $\phi ! x$ would be predicative.)
  It thus appears that $\phi ! x$ fails to be predicative just as much by
  the terms it does not include as by the terms it does.  Again, given
  any class $u$, the property $\phi ! x$ belongs either to some some, but not all, of
  the members of $u$ or to some, but not all, of the members of 
  not-$u$.  This is the zigzag property which gives its name to the theory 
  we are considering.

Russell goes on to explain a bit more.

The zigzag theory, in some form or other, is that assumed in the
  definitions of cardinal and ordinal numbers as classes of classes (if
  numbers are supposed to be entities). For all these classes of classes, if
  they are legitimate, must contain as many members as there are entities
  altogether; hence, if bigness makes classes go wrong, as we suppose in
  the "limitation of size" theory, cardinals and ordinals so defined will
  be illegitimate classes.

A: As a footnote to Arthur Fischer, here's an additional quote from Michael Potter's Set Theory and its Philosophy:

In 1906, Russell canvassed three forms a solution to the paradoxes might take:
  the no-class theory, limitation of size, and the zigzag theory. It is striking that a century later all of the theories that have been studied in any detail are recognizably descendants of one or other of these. Russell’s no-class theory became the theory of types, and the idea that the iterative conception is interpretable as a cumulative version of the theory of types was explained with great clarity by Gödel in a lecture he gave in 1933, ... although the view that it is an independently motivated notion rather than a device to make the theory more susceptible to metamathematical investigation is hard to find in print before Gödel 1947. The doctrine of limitation of size ... has received rather
  less philosophical attention, but the cumulatively detailed analysis in Hallett
  1984 can be recommended. The principal modern descendants of Russell’s
  zigzag theory -- the idea that a property is collectivizing provided that its syntactic expression is not too complex -- are Quine’s two theories NF and ML.
  Research into their properties has always been a minority sport: for the current
  state of knowledge consult Forster 1995. What remains elusive is a proof
  of the consistency of NF relative to ZF or any of its common strengthenings.

