[FOM] Axiom schemata

[Andreas Blass]

	Matt Frank asked "Given that we generally prefer finitely
axiomatized theories to infinitely axiomatized theories, why do we tend to
use ZF instead of NBG?"  Several people have already commented that the
"given"  part of the question is not so obvious.  It is not difficult to
set up a deductive system in which schemas can be handled just like single
axioms.  For example, Morse's book "A Theory of Sets" (which, despite its
title, presents a version of Morse-Kelley class theory) sets up a logical
framework of this sort.
	As for the other part of Matt's question, a lot depends on whom
"we" refers to.  Mathematicians generally have no need for proper classes,
don't care much about axioms, and will answer "ZF(C)" when asked for the
foundation of their work only because that's what they've been trained to
answer.  So I'll interpret "we" as "set theorists" or, better yet, as "set
theorists who agree with me."  In other words, why do I use ZF rather than
NBG?
	In the first place, although I say that I use ZFC and not NBG,
it's not entirely clear that I'm telling the truth.  I certainly say
things like "suppose V=L" or "let j: V ---> M be an elementary embedding",
which, on the face of it, are about proper classes.  I like to imagine
that my talk about classes is just an abbreviation of talk about sets, as
explained for example in the early part of Jensen's book "Modelle der
Mengenlehre".  That idea covers "V=L" but a bit more work is needed to
cover elementary embeddings --- they would need to be definable from sets,
and sometimes I don't want to require that.
	So the question has become: Why do I claim to use ZF rather than
NBG?  The main reason is that I have a better intuition of the ZF universe
--- the cumulative hierarchy --- than of the NBG universe.  That may seem
strange since, given the cumulative hierarchy of ZF, I could just add one
more level and get a perfectly good universe satisfying NBG (or even MK).
The problem is that part of my intuition of the ZF universe is that it
continues "forever".  If there were another level, the level of proper
classes, then the ZF hierarchy should have continued, to include that
level and many more beyond it.  To put it another way, class theories like
NBG and MK seem unnatural because they say there's a level containing
proper classes but we can't form another level beyond that.
	Why doesn't finite axiomatizability influence me in favor of NBG?
Partly because, as indicated above, I don't see much harm in allowing
schemas.  But there's another reason too.  The most "natural"
axiomatization of NBG has a comprehension axiom scheme saying that you can
form the class of all sets x that satisfy A(x) provided each quantified
variable in A(x) ranges only over sets, not over proper classes.  That's
an infinite number of axioms.  Finite axiomatizability is achieved by
showing that all these axioms follow from a finite number of them ---
essentially by using G"odel's operations to simulate the syntactic
construction of first-order formulas.  The price of finite
axiomatizability is (in my opinion) a great loss of naturality.  Even if I
were to use NBG as a foundation for something, I'd be inclined to use the
infinite but natural-looking axiomatization rather than the finite but
ad-hoc-looking axiomatization.

Andreas Blass