A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Sat Jul 17 03:26:39 EDT 2004

   Neil Tennant SEEMED to be asking (see below) two questions, one a 
question of fact about mathematical practice, one a question about 
its justification.
   Superficially, the question of fact seems to be hands-down in favor 
of ZF (Lemmon's little "Monographs in Modern Logic" book and the set 
theory chapter of Mendelson's "Mathematical Logic" are the only basic 
introductions to set theory I know that use NBG, as opposed to ... 
how many? ... that use ZF!).  A bit below the surface it's not so 
clear.  We don't SEEM to think of the axioms of ZF as forming an 
unstructured set: as Kreisel pointed out
   	"Bourbaki is extremely careful to isolate the assumptions
	of a mathematical theorem, but never ...what instances of
	the comprehension axiom are used.  This practice is quite
	consistent with the assumption that what one has in mind
	when following Bourbaki's proofs is the second-order axiom,
	and the practice would be horribly unscientific if one
	really took the restricted schema as basic."
[Kreisel, "Informal Rigour and Completeness Proofs," in Lakatos, ed., 
"Problems in the Philosophy of Mathematics" (1967); quoted sentence 
is on page 88 of te reprinting in J. Hintikka, ed., "The Philosophy 
of Mathematics."]

    Again, many writers who are "officially" using ZF refer freely, in 
the natural-language part of their expositions, to classes.  Teakeuti 
and Zaring's textbook is unusually careful, but otherwise typical, in 
giving a preliminary account of how this can be interpreted as a 
façon de parler for discussing formulas.  [I owe the observation 
about T. and Z.'s book to Charles Parsons's "Sets and Classes," in 
"Nous" v. 8 (1974); repr. in Parsons's "Mathematics in Philosophy." 
I think Parsons's paper is one of the best and most useful 
discussions of the set/class issueavailable.]  Since the 
class-existence axioms of the finite axiomatization of NBG are 
designed to mirror the inductive definition of formula-hood, this 
means that what they actually WRITE reads like a sequence of proofs 
in NBG!

    My own-- fairly trite-- view is that the primary reason for 
choosing ZF as one's "official" axiomatization is ontological. 
Either (proper) classes are the same sort of entities as sets, or 
they are fundamentally different, to be reasoned about in a different 
and more constructive way.  On the first alternative (i) the natural 
system is not NBG but the stronger MK, but (ii) it seems bizarre to 
postulate a single, additional, "layer" of the things on top of the 
topless iterative hierarchy.  But the best understood implementation 
of the second alternative is to take classes as conceptual or 
linguistic in nature, and on this view it seems more elegant to 
"adopt" a formal language whose variables range only over sets, 
keeping talk of classes in one's more orless informal, more or less 
rigorously discussed, metalanguage.
>On Mon, 12 Jul 2004, Matthew Frank wrote:
>>  Given that we generally prefer finitely axiomatized theories to infinitely
>>  axiomatized theories, why do we tend to use ZF instead of NBG?  --Matt
>Do we, really? And if we don't, why should we?
>Neil Tennant
Allen Hazen
Philosophy Department
University of Melbourne
(currently trying to figure out what to say next term in an undergraduate
unit to which my head of department has given the name "Philosophy of Logic.")

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