What are the primitive notions of the standard axiomatizations of set theory (ZF, Goedel-Bernays, Quine's...

In article <3830@garfield.UUCP>, robertj@garfield writes:
>                              the Fraenkel-Zermelo [Editor: "Frankel-Zermelo"
>in the original] Axiomatization in which
>class and membership are the primitive (undefined) concepts (not sets!)
>and a set is defined as a class which is a *member* of another class.

In point of fact, in ZF sets are the primitive notion, and classes are
just a convenient way of talking about propositions.  For example, one
has in ZF that `{x|phi(x)}={x|psi(x)}' is just an "abbreviation" for
`for all x, phi(x) iff psi(x)', and similarly for some other basic no-
tions involving classes, like `y in {x|phi(x)}', meaning `phi(y)'.  One
does not define `{x|phi(x)} in y'.

In contrast, in GB (Goedel-Bernays), classes are primitive, and are much
as you describe.  ZF and GB are essentially the same.  There is a stronger
theory KM (Kelley-Morse) which gives more life to classes.

-- Matthew P Wiener <ucbvax!brahms!weemba>

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[ Editor: posted by Dale Worley <cullvax!drw@eddie.mit.edu> for Jack
Campin, whose news software could not post outside Scotland. ]

The original statement of NF is Quine's 1936 "New foundations for mathematical
logic", reprinted in his collection "From a Logical Point of View". The axioms
are incredibly simple - just extensionality and a universal comprehension
axiom for "stratified" predicates, those where all variables can be assigned
a type (an integer) under the constraint: if "x is an element of y" occurs in
the predicate, then the type of y must be one higher than the type of x. This
rules out all the obvious paradoxes, since e.g. "x is not an element of x" is
not stratified. But it immediately gives you a universal set, which can be
shown to be a member of itself. Sets corresponding to nonstratified predicates
can certainly exist, their existence just doesn't follow immediately from the
comprehension principle - for example, the axiom of infinity isn't stratified.
There is lots of fun to be had trying to get stratified definitions of things
we take for granted in other set theories.

Rosser's "Logic for Mathematicians" attempts to use it in anger, Principia
style - as mathematical induction is independent of NF, he had to extend it
with an "axiom of counting" to get the natural numbers.

The best introduction to NF I know of is a chapter in William S. Hatcher's
"The Foundations of Mathematics" (a superb survey of the whole field).

For the state of the art, there is a monograph by Thomas Forster: "Quine's NF:
an introduction", Cahiers du Centre de Logique no. 5, CABAY, Institut
Superieur de Philosophie, Universite Catholique de Louvain, Louvain, Belgium.
Be warned, it's hard going. There's a review of it in JSL vol. 50 # 2 (1985).

Papers on NF appear sporadically in the Journal of Symbolic Logic, usually by
Boffa, Henson, or Forster: the most recent one I've seen is Forster's "The
status of the axiom of choice in set theory with a universal set", JSL vol. 50
no. 3 (1985). You can follow pointers back from there.

NF may well come in handy some day for describing polymorphic type systems
where types are objects. The BIG problem with NF that it shares with these
type systems, and that makes it hard for most set theorists to accept, is that
there is no known model of it in a conventional Zermelo-style cumulative type
system - there are some partial results, the most illuminating being one by
Jensen that NF can be modelled in ZF if extensionality is weakened to allow
for urelements (this system is called NFU).

-- Jack Campin <jack@cs.glasgow.ac.uk>

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In article <682@uwm.edu> litow@csd4.csd.uwm.edu (Bruce E Litow) writes:
>Is there a finite axiomatization of set theory ?

Depends on what you mean by "set theory".  If you're asking "Can ZFC
be finitely axiomatized?", the answer is that it can be finitely
axiomatized if and only if it is inconsistent (so we'd better hope
that it can't be finitely axiomatized).  If, on the other hand, you're
asking "Is there some set theory with the same `power' as ZFC which is
finitely axiomatized?", the answer is "yes".  Goedel-Von Neumann-Bernays
set theory is a conservative extension of ZFC which is finitely
axiomatizable.

-- Ian Sutherland <ian%oravax.uucp@cu-arpa.cs.cornell.edu>

No and yes.  The "standard" set theory, ZF, has no finite axiomatization.
(See Kunen's book on set theory.)  It has a conservative extension, NBG
(with class variables) that is finitely axiomatizable.

What is less well known is a general theorem in logic (due to Kleene)
about converting any recursively axiomatizable theory to a finitely
axiomatizable one, by expanding the language to include additional
predicates.  But sci.math is not the place for that.

-- Herb Enderton <hbe@math.ucla.edu>