[FOM] predicative foundations
Nik Weaver
nweaver at math.wustl.edu
Wed Feb 15 16:36:05 EST 2006
There are strong philosophical reasons for adopting a
predicativist foundational stance, which I have discussed
in some detail in a number of messages I've posted on this
list over the past several months. Much of what I've had
to say involved criticism of the idea that there objectively
exists a unique well-defined metaphysical world of sets. If
one accepts such a notion then predicativist critiques have
little force. But one then has to deal with the classical
set-theoretic paradoxes --- in particular, if this supposed
universe is completely well-defined then presumably we should
be able to talk about the set of all entities lying in it,
and that seems to lead directly to contradiction. A related
difficulty is that this kind of platonistic view on its face
seems to justify unrestricted comprehension, again leading to
paradox. One can try to avoid these difficulties by conceiving
of the set-theoretic universe as an incomplete entity which is
in some sort of "perpetual, atemporal process of becoming"
(Maddy). The retort would then be that this notion is simply
incoherent.
The preceding arguments are indirect: if one accepts platonism
then various difficulties arise, therefore doubt is supposed to
be cast on the platonistic approach. It is also possible to
directly criticize the belief that when we talk about sets we are
actually referring to a certain kind of well-defined abstract
entity, by arguing that "the very notion of a set ... is based
on a series of grammatical confusions." This quote is taken from
the paper "Grammar and sets" by Hartley Slater, to appear in the
Australasian Journal of Philosophy. Section 2 of that paper
contains a thorough, and in my opinion, absolutely decisive
refutation of the platonic conception of sets.
Anyone who accepts a platonic foundational stance can comfortably
dismiss the predicativist contention that sets must be "built up
from below". After all, if sets are simply "there" then we have
no particular reason to believe that they all must be reachable
from below, definable in a non-circular manner, etc. Conversely,
if one rejects platonic ideas then it becomes very difficult to
maintain this attitude.
Harvey Friedman blandly asserts that "claims that predicativity
has some special place in the robust hierarchy of logical strengths
ranging from EFA through j:V into V are unjustified." Can he give
a compelling justification for any impredicative system that does
not assume a platonistic conception of sets?
In his previous message in this thread he claims that ZFC can be
justified by "extrapolation from finite set theory". That is a
good example of an explanation that I would not consider compelling.
He also writes "There is also a clear philosophical basis for the
impredicative comprehension axiom scheme" but does not tell us
what it is.
Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
http://www.math.wustl.edu/~nweaver
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