Willard Van Orman Quine

272 joseph s. ullian
Quine’s early work in logic and foundations grew out of his focus
on Principia Mathematica. 2 He sharpened that work’s apparatus and
recommended improvements, ever with an eye to conceptual, notational,
and ontological economies. This was what his dissertation
and resultant first book were about. His work in set theory reached a
high point with the system ML of his book Mathematical Logic.ML
is the impredicative enlargement of an earlier Quine system, the NF
of his paper “New Foundations for Mathematical Logic.” In each the
provision for set existence adopts a central idea of Principia, where
paradox was avoided by requiring a set to have “higher type” than
its members. ML’s sets are just those of NF, but ML allows additionally
for the existence of classes that are not sets, classes that do
not themselves bear the membership relation. Sets are all classes,
but not all classes are sets. So ML has a “two-sorted” universe. ML’s
extra level, allowing broader scope to its quantifiers, enables sharpened
definitions and a development somewhat smoother than that
of NF.
It was in ML that Quine placed his hope. Though born of the typetheoretic
insight, it lacked type theory’s ugly encumbrances; it was
stark and elegant in its formulation. But ML has been plagued by
problems, and it has not gained favor among set theorists. Early on,
Rosser and Lyndon discovered an inconsistency in the system; Wang
offered a repair that was in fact a gain in elegance as well. Deep problems
with ML’s ordinals became apparent. But worse, it was found
that the status of the class N of natural numbers, of all things, was in
question. You would want N to be a set; the development of standard
mathematics demands it. Yet it was Rosser again who showed that
set-hood of N is unprovable in ML, unless ML is inconsistent.
Quine’s systems of set theory are seen by most as technical curiosities
rather than as serious contenders for adoption. While ZF,
the reigning theory, has thrived as the whole subject has flourished,
ML and NF are charged with not capturing the basic intuitions or
insights about the realm of sets as an iterative hierarchy. The current
attitude is that formal adequacy of a set theory is not enough;
reflection of how the realm is to be conceived is wanted as well.
Quine, no champion of intuition in the foundations of mathematics,
has had other goals: In Dreben’s phrase, he has undertaken “syntactic
exploration” in quest of an appealing apparatus strong enough
to float standard mathematics while guarded enough to keep off the
reefs of paradox.
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