[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:
Koskensilta:
"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."
Jones:
"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
classes?
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
variables.)
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
carefully.)
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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