FOM: are sets made up of their elements?
holmes@catseye.idbsu.edu
holmes at catseye.idbsu.edu
Tue Jun 6 11:04:58 EDT 2000
Dear FOMers,
More on the question of whether sets are made of their elements. I
reproduce an insertion from my last note:
Simpson said (in reply to text of mine also included):
> I think that many users of set theory (I hope not any set
> theorists!) actually think on some unreflective level that the set
> is somehow "made up" of its elements [...]
I guess I am one of those users, because I certainly do think that a
set is made up of its elements. What is wrong with this view? Why
does Holmes say that it is not based on reflection?
I continue:
It is fair to ask for an alternative view of how sets are "made" if
they are not made of their elements, and I will provide an answer (or
more than one possible answer). It suits my purposes here to concern
myself entirely with ZFC foundations -- the issue here has little to
do with choice of set theories (though a proper understanding of the
notion of set may help us to understand that there _is_ a choice of
set theories...)
I reproduce another remark of Simpson's (re set theory with AFA):
>Proponents of AFA may want to refer to the elements of V* as ``sets''.
>But this entails a massive revision of standard terminology, and I see
>no good reason for it.
Simpson appeals to history (Brouwer's recorded views) in making his
assessment of the philosophical motivation of constructivism. I appeal
to history here against his view that the ZFC "sets" are the only
sets which can be called sets. The Axiom of Foundation was not part
of Zermelo's original formulation of his set theory. Neither was
full extensionality (urelements were allowed). Early workers in set
theory did consider the possibility of non-well-founed sets. New
Foundations and NFU are set theories. In short, the axiom of foundation
is an axiom proposed and now generally adopted for set theory, not
part of the informal definition of what a set is...
That left aside, I proceed to whether sets can coherently be regarded
as being made up of their elements.
1. They clearly can't. Otherwise there is no way to distinguish between
{x} and x.
2. Nonetheless, almost everyone is strongly tempted to say that they
are. I think the reason for this is that no coherent alternative
account is usually given. As I noted earlier, the idea of a set being
made up of its elements actually works as long as one is talking about
point sets (it works on the Venn diagram level). If you "picture" the
elements of your sets as pairwise non-overlapping objects, then there
will be a one-to-one correspondence between the sets of these
non-overlapping elements and the composite objects made up of these
elements as parts. Everyone sees an informal account of sets along
these lines while they are learning about sets -- and perhaps sees no
other informal account of what a set is at all (just axioms of a formal
system from then on...) I think this is why people tend to think this
way.
3. As soon as you try to construct sets of sets, this breaks down.
If you think that {1,2}, {2,3}, and {1,3} are composites made by
putting numbers together, and apply the same intuition to {{1,2},{2,3}},
you will find it hard to explain the difference between {{1,2},{2,3}}
and {1,2,3} (in terms of that conception -- of course all of us can
say why they are different, but this understanding does not come from
the idea that sets are composites).
4. So now I'll present my analysis (one possible analysis at least).
I will need to draw a set/class distinction. I identify classes with
_properties_: in fact, classes are the referents of the second-order
variables in second order logic... Classes are _not_ sets -- sets are
objects and classes are not. Classes _are_ determined by their
elements (and nothing else), but they don't have their elements as
parts. But classes are not sets -- a class can only have objects as
its elements, not other classes. (one could introduce classes of
classes and go on toward the theory of types, but I won't).
5. Sets differ from classes in being objects themselves and so being
able to be themselves members of classes and other sets. A set is
determined when we assign a class as extension to an object. Moreover
(and I regard this as part of the definition of the set concept) we
will not assign the same class as extension to more than one set
(though there may be objects to which we assign no extension at all,
which will have the same elements as the empty _set_ to which the
empty class is assigned).
6. I claim that a set theory is determined by any injective scheme
for associating classes with objects. A set has as "ingredients"
not only the objects in its extension, but also the object that it
_is_ -- and that is the crucial extra ingredient.
Notice that nothing about this picture either encourages or discourages
the presence of non-well-founded sets. An object can be assigned an
extension which contains that same object...
(an extra ingredient which can reasonably be added to this picture (as
David Lewis has suggested) is to make the relation of part and whole
on sets coincide with the subset relation -- if we assume the
existence of a domain of pairwise non-overlapping objects which we can
use as "singletons", we can build composite objects corresponding to
every class of those objects to which we have assigned singletons
(only objects with singletons are "sets" in this version), so we get
the idea that sets are "made up" of the singletons of their elements
-- but this is really just a refinement of the essential point which
we will leave aside)
7. The iterative conception of set makes perfect sense in this light,
except that its weak point (it _does_ have one) is made a little clearer.
We begin with nothing. We consider the empty class and assign it to
an object (where did that object come from?) We proceed through
ordinal stages and at each stage take all subclasses of the objects
constructed at previous stages and assign each subclass as an
extension to a new object (a lot of new objects are coming from
somewhere!). This appears to be a coherent scheme (there is nothing
obviously contradictory about it!); the metaphysical difficulty which
is glossed over is whether there are actually enough objects to
implement it. Usual accounts of the iterative hierarchy invite us to
believe that we get the new sets "for free" -- this is an "advantage"
of the view that sets are determined by their elements, and only by
their elements. It has all the advantages (as Russell said in some
similar context) of theft over honest toil...
The "weak point" is that ZFC assumes that the universe is _very large_.
We all know this, but we ought to think about it carefully. I don't
regard this as a fatal defect (and I should note that NFU + Infinity
also supposes a quite large universe, and my favorite extensions
of NFU suppose a very large universe indeed).
8. There's another nice account of the iterative conception of set
which I'd like to outline. This uses second-order logic explicitly.
We consider a universe of objects (about which we make no assumptions
yet) and another sort of binary relations on those objects (something
we can do in second-order logic). In this account, sets aren't
treated as objects at all. (we can also use a sort of classes, but
classes are easily coded as binary relations).
Sets are interpreted as isomorphism classes of well-founded
extensional relations with a top element. We can't really build the
isomorphism classes (we have no sort of sets of binary relations), but
we don't need to -- we can define the notion of two well-founded
extensional relations with top element being isomorphic, and use that
to interpret equality of sets.
There is a natural notion of "membership" between isomorphism classes
which can be used to interpret membership in the implemented set
theory.
The resulting "set theory" will not in any case be ZFC (this requires
a refinement). Like the previous analysis, this brings out the fact
that we must assume that there are lots of objects.
With no axioms of infinity, we get very little: extensionality, foundation,
separation, replacement, and union will hold (given the obvious comprehension
axiom for binary relations).
If we assume that there are infinitely many objects, we get pairing
and infinity as well. We also assume that the universe can be
well-ordered.
Power set is demonstrably false in the scheme as I have described it so
far! The problem is that there will be "sets" of the size of the universe
of objects, and they will not have power sets! (without choice, the
situation is more complicated, but power set is still false).
So we make an additional restriction -- we require that the
well-founded extensional relations with top also have "domain" smaller
than the universe of objects. At this point we lose the axiom of
union unless we assume (in effect) that the cardinality of the
universe of objects is regular, and we can have the axiom of power set
if we assume that the cardinality of the universe is strong limit.
This analysis ends up with ZFC, exactly, (it is first-order ZFC, even
-- we do not have a second-order theory of classes!) but once again
brings out the point that we need to assume the existence of a lot of
objects.
This is basically the way that ZFC - Foundation + AFA is interpreted
(though the metatheory is set theory rather than second-order logic as
here) -- if we look at different (super) classes of relations which
include some non-well-founded relations we get different set theories.
There is no fundamental conceptual difference between ZFC and ZFC -
Foundation + AFA when they are presented in this format, though ZFC is
clearly simpler! Here the "sets" of ZFC are identified with relations
that "picture" their transitive closures (up to isomorphism).
9. I hope that it is abundantly clear that I am NOT criticizing ZFC
foundations in any way, shape or form in this posting, and that my
critique of the notion that sets are made up of their elements has
nothing to do with a program to overthrow Zermelo and Fraenkel :-)
(These concepts can also be used in an analysis of NF or NFU, and in
fact can be used to demonstrate that stratified comprehension makes
sense on semantic grounds as well as at the level of a "syntactical
trick". But it takes a bit more work to give an account of NFU in
these terms than it does to give an account of ZFC, and this is not my
aim here.)
P.S. I notice a point to which I didn't respond in Simpson's post--
the "technical errors" to which I alluded earlier (caused by the
notion that sets are made up of their elements) are ones certainly not
made by set theorists: I was thinking of errors typically made by
undergraduate students learning about the set concept.
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes
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