[FOM] The defence of well-founded set theory

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Thu Oct 6 02:14:46 EDT 2005

    Aatu Koskensilta and Roger Bishop Jones, arguing about the status of NBG:
	"NBG is consistent with the usual conception of set theoretic
	hierarchy in a rather trivial sense: one can consider the
	totality of classes to consist of properties or collections
	definable in the language of set theory. The intelligibility
	of the language of set theory itself seems to imply the
	acceptability of such a totality of classes."
	"I find this hard to swallow.
	The iterative conception of set describes the pure
	well-founded sets, which we may accept as the domain
	of discourse of first order set theory (ZFC) [...]
	However, as soon as we try to make the totality of
	pure well-founded sets into a collection we run into
	trouble, because of course it must be and cannot be
	a pure well-founded set."
--To which I  remark that this is a familiar objection to set 
theories admitting "proper" (or, in Quine-speak, "ultimate") classes. 
Put in Platonistic language: the ***intended interpretation*** of 
ZFC had the  variables ranging over ALL THE SETS THERE ARE, so what 
are you doing postulating ADDITIONAL classes?  And surely there is no 
plausible reason to suppose the set-theoretic hierarchy isn't 
topless, so what's the rationale for postulating a "top" level of 

    Jones immediately continues:
	"Calling this totality a class rather than a set doesn't
	really help, because if there were a class which collected
	ALL pure well-founded sets then it would be a pure
	well-founded collection and there would have to be a set
	with the same extension."
---To which I remark that this would be a telling objection ***IF*** 
we assumed that sets and classes were entities of the same kind, but 
we don't have to assume that.  Sets are mathematical objects, things 
of a sort not dependent on any sort of conceptualizing (as witness 
the fact that there are sets not definable in any reasonable 
extension of a language we conceptualizers can understand).  Classes, 
on the other hand, are conceptual: they are the (extensionalizations 
of) meanings of predicates of our set-theoretic language, and they 
exist only by being definable.  (For the distinction, cf. Charles 
Parsons's  "Sets and classes" ["Nous," 1974; repr. in Parsons's 
"Mathematics in Philosophy" (which, by the way has recently been 
re-issued in paperback!!!!)].  And, to prevent misunderstanding, note 
(what Parsons pointed out elsewhere) the definitions may be 
parametric: NBG allows classes defined by formulas containing free 
      To put it another way: suppose you think "the" sets in "the" 
cumulative hierarchy are a definite enough universe to reason about. 
You can construct a theory about them in a First-Order language: 
that's ZFC.  You could be more liberal, linguistically, and formulate 
your theory in a stronger language, say Ramified Second-Order Logic 
with only one level of ramification (what Church, in  section 58 of 
his 1956, calls the "Predicative" calculus of Second-Order logic). 
This, with inessential changes of terminology and notation, is NGB.

    Koskensilta continued:
	"The only controversial thing in NBG not motivable in
	this fashion is the global axiom of choice for which you
	need to use forcing to prove conservativity."
(Comment: I think this might, pedagogically, be a good FIRST topic in 
introducing forcing: forcing conditions are simple (just subsets of 
the class graph of the selection function), and I don't think you 
need to invoke generics: if we take the forcing conditions as the 
"worlds" of a Kripke-model, we validate  an intuitionistic theory 
classically equivalent to NBG.  But I haven't checked the details 

    Jones replied:
	"But surely choice is (not withstanding Boolos' reservations)
	entailed by the requirement in the iterative conception that
	at each stage ALL sets are formed whose members have been
	formed at previous stages."
---Two comments:
	On the matter at hand, NBG, the issue concerns GLOBAL choice: 
just reformulating ZFC with Predicative Second-Order Logic would 
leave you with choice for sets (the C in ZFC), but allow there to be 
CLASSES of disjoint non-empty sets with no choice class.  Global 
choice is problematic from the standpoint of the predicative logic, 
because it is the one  point at which the usual NBG  axioms 
postulate a class (the graph of a global selection operator, or 
something equivalent) which is NOT definable (even parametrically) in 
the language of set theory.
	On Boolos's reservations: it seems to me that to get the Set 
Theoretic axiom of choice out of his sets-and-stages conception you 
have to appeal to something like a choice principle in the 
sets-and-stages theory: for any bunch of disjoint sets formed at some 
stage, there will  be a choice set formed at the same stage.  You get 
out what you put in.

Allen Hazen
Philosophy Department
University of Melbourne

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