[FOM] Sets and Proper Classes
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Wed Oct 29 04:55:40 EST 2003
Dmytro Taranovsky wrote:
>The purpose of this posting is to correct some misunderstandings about
>the term "proper class".
>
>A set is an arbitrary collection of objects. A property of sets is a
>formula with parameters and one free variable such that a set either
>satisfies the formula or it does not. It is convenient to view
>properties as collections, where x in {y: phi(y)} means phi(x) and {x:
>phi(x)}={y:psi(y)} means for all x (phi(x) iff psi(x)). However, there
>is no collection of all collections, so for not every formula is there a
>collection of all objects that satisfy the formula. A property of sets
>for which there is no set of all sets that satisfy the property is known
>as a proper class.
>
>We know that for every set s, there is the set of all subsets of s. The
>same reasoning should lead to that for every collection s there is the
>collection of all subcollections of s. Therefore, if proper classes did
>exist as collections, then we could form a superclass of all classes of
>ordinals, proceed to form a super-superclass of all sub-superclasses of
>the superclass of all classes of ordinals, and so on. In that case, by a
>set we meant a set in the initial segment of the cumulative hierarchy,
>perhaps, a member of V(kappa) where kappa is the least inaccessible
>cardinal such that V is an elementary extension of V(kappa) with respect
>to first order formulas about sets, and by a class, we meant a subset of
>V(kappa).
>
>
Ah, but there's so much more to proper classes than this!
The problem with ordinary theories of proper classes, from a certain
point of
view about which I say a bit more below, is that there is no sensible
distinction
between proper classes and sets: it seems that we have just forgot to
add one
stage to the cumulative hierarchy, this stage comprising the proper classes
which are viewed merely as quasi-combinatorical collections (i.e. exactly
as sets are) of the universe. This is mirrored in the natural models of
all classical theories of classes which are of form V(\alpha + \beta),
where \beta
is a small ordinal (1 for Morse-Kelley and omega for Ackermann style
theories, if I recall correctly).
This is in drastic contrast to the usual informal exposition of the
cumulative
hierarchy going "as far as possible" with the quasi-combinatorical
collecting
of elements into sets at stages. According to this picture there simply are
no proper classes, since if there were, they would be "potentially existing"
quasi-combinatorical collections of sets, but all of these are supposed to
be exhausted in the "process" of making sets.
There have been various attempts to give proper classes (and classes in
general)
a distinct identity differing from that of sets. For example, there is
the theory
of shifting sets, in which the epsilon predicate is ternary, the middle
argument
being the "stage" in which membership to the last element is considered.
Proper
classes are "incompletable" according to this theory in the sense that
there is
no stage in which their membership would become entirely fixed. This has
been
also informally suggested as the defining property of proper classes: whilst
a set retains its extensional identity if we enlarge a natural model of set
theory, proper classes don't. This is also closely tied to the idea that the
identity of a proper class is defined by some formula of which it's the
extension.
There have also been suggestions by Penelope Maddy and others such as
Parsons
to take seriously the idea that proper classes *are* predicates
considered extensionally.
Of course, Russell's paradox shows that this can't work in its simplest
form, but the reason
for this failure is clear: if we consider membership in class to be
satisfaction for the
predicate the class is the extension of, and spell this out, we actually
get the liar
paradox from Russell's paradox. There have been various solutions to the
liar
paradox, the first being the Tarski hierarchy of meta-languages. This
would give
us a hierarchy of classes. Another is the Kripke fixed point
construction, according
to which neither {x | ~x\in x} \in {x | ~x \in x} nor ~({x | ~x\in x}
\in {x | ~x \in x}).
If we build a Tarskian hierarchy on top of the Kripke fixed point, we do
get
{x | ~x\in x} \in_1 {x | ~x \in x}, and we can go on, defining in effect
a ternary
membership relation \in_alpha (for all ordinals alpha), which is related to
the ternary relation of the shifting sets.
There is a problem with theories of classes which are based on
satisfaction or
truth predicates, namely the very close tying of classes to formulae of the
language. Why should classes be tied to a particular first order
language? Why
not allow for infinitary languages, or even second order languages?
Also, Lear has suggested in somewhat Zermelian spirit as a model for set
theoretic
practice a Kripke model, i.e. a partially ordered set of stages of
development, which
are taken to be partial approximations to the real set theoretic
universe, so that
proper classes of each stage are sets of some later stage. This sort of
idea naturally
leads one to consider adopting intuitionistic logic for collections
which are not
sets, since they are the Cantorian counterparts of the potentially
infinite collections,
"continously coming into being".
--
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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