FOM: some comments and questions, and the nature of the set concept
Randall Holmes
holmes at catseye.idbsu.edu
Mon Jan 26 14:26:48 EST 1998
This is from M. Randall Holmes.
part 1 (comments on Friedman reply to an earlier posting)
(quote from Friedman's reply to an earlier posting of mine)
This post is inspired by Randall Holmes, 2:58PM 1/23/98.
A common fatuous objection to set theoretic foundations of mathematics is
that it picks ad hoc presentations of basic structures in mathematics and
saddles mathematics with ridiculous "theorems" like the intersection of e
and pi is e. Or some ad hoc definition of pairing like (x.y) = {{x},{x,y}}.
(end quote)
I hope that it is clear from my original posting that I do not have
such objections. Friedman admits to being inspired by my remarks but
doesn't say anything about them directly.
(further quote from the same Friedman posting)
Fatuous objections like these are uniformly handled in the manner indicated
by my positive posting 5:42PM 11/15/97, 5:Constructions. Applied to here,
we use the standard set theoretic development to prove what might be called
"literal" R, say by Dedekind cuts, with its ordered field structure. This
is what is riddled with phenomenon like the intersection of e and pi. Then
we have proved "there exists a complete ordered field." Then we apply the
rule in the previous posting that asserts that we can introduce a constant
symbol for a complete ordered field. Then there is no more problem of the
kind the other side is complaining about. In practice, of course, one
introduces a complex of constants for the components of an ordered field.
(end quote)
This is an entirely adequate practical solution to the problem
referred to. It is interesting to observe that since these new
symbols are (presumably) being introduced in the context of ZFC, the
elements of one's new complete ordered field (e and pi, for example)
are still provably sets (since everything in the ZFC world is a set)
and so e and pi _do_ have an intersection -- but one cannot prove any
theorems about this intersection because one has no information about
which sets e and pi happen to be. This logical move is an exact
analogue of defining an abstract data type and hiding the details of
its implementation.
part 2 (a question for Friedman)
I have been looking at the paper on transfer principles on your web
site, and I notice that it is part of your program to show that the
falsehood of the continuum hypothesis follows from such transfer
principles. Do you actually have a result with ~CH following from a
transfer principle?
part 3 (the intuitive notion of collection versus the mathematical
notion of set)
The question of the relationship between the intuitive notion of
"collection" and the mathematical notion of "set" has been raised
several times. I'm going to argue here that these notions are
actually rather different, and that there was a certain amount of
intellectual sleight-of-hand involved in replacing the intuitive
notion of "collection" with the new mathematical notion (and there is
still sleight-of-hand involved in the teaching of this concept).
Ideas along the same lines are found in the works of the philosopher
David Lewis (particularly his book _Parts of Classes_, although the
development here is independent of his.
The difference between the intuitive notion of collection and the
mathematical notion is that "collections" in everyday experience are
wholes of which their "members" are non-overlapping parts. It is
impossible to understand the sets of ZFC in this way, but one finds
philosophers (more understandably) and mathematicians (who should know
better) using language which suggests that our intuitive notion of
collection is the basis of the mathematical idea.
Halmos, _Naive Set Theory_, p. 1: "a pack of wolves, a bunch of
grapes, or a flock of pigeons are all examples of sets of things".
All of these share the characteristics noted above: a pack of wolves
can be understood as a (disconnected) physical object whose parts are
the individual wolves (which we can rely on not to overlap with each
other). On this interpretation, a "pack" of one wolf (a curious idea
from the commonsense standpoint) is simply to be identified with that
wolf. This is the crucial point where the mathematical notion of
"set" breaks with the dominant everyday sense of "collection".
The crucial thing is the difference between the relation of part to
whole (the study of which is called "mereology") and the relation of
element to (mathematical) set. It is quite clear that the notion of
"part" appropriate to sets is _not_ "element". The relation of part
to whole is transitive, and the relation of set to element is not. A
transitive relation which can plausibly be understood as the relation
of part to whole as restricted to sets is the relation of inclusion: a
part of a set is a _subset_ of the set, not an _element_ of the set.
If we understand the inclusion relation as the relation of part to
whole, then we _can_ understand a set as a "collection" in the
intuitive sense outlined above -- with a twist. A set is a whole made
up of distinguished non-overlapping parts -- its one-element subsets.
These distinguished parts are related to its elements, but they are
not identical with its elements.
Understanding the element-to-set relation reduces in this way to
understanding what the relation of x to {x} is, if we allow ourselves
to understand the inclusion relation as the relation of part to whole:
x \in y is equivalent to {x} \subseteq y
Lewis gets this far, but treats the relation of a singleton set to
its element as a mystery. I don't think that this is necessary; I think
that a little thought can show how we are to understand the singleton
construction, and what it does for us.
Let's go back to a real world example. Suppose that one is
considering a committee with four members, and one asks how many ways
one can form a subcommittee owith two members from this committee (the
answer, of course, is that there are 6 committee rosters). Suppose
that the members of the committee are A, B, C, and D. Each
subcommittee can be understood in the usual intuitive way as a
disconnected physical object: AB, for example, has A and B as parts.
Now suppose that we wish to consider the possible subcommittees with 3
members; there are 4 of these. Now impose the additional restriction
that A and B cannot both be members (A and B fight like cats and
dogs): there are only 2 of these. What is the difference between the
set {ABC,ACD,BCD,ABD} of all 3-member committees and the set {ACD,BCD}
of committees on which A and B do not both serve? If we tried to form
them along the lines of the intuitive conception, both would turn out
to be the physical object ABCD which has all four committee members as
parts. The difficulty here is that subcommittee rosters do not fulfil
our expectations of members of collections; they are not
non-overlapping, and, even worse, a collection of some of them can
include the whole of some other one of them. This illustrates why the
intuitive notion of collection does not support membership of sets in
further sets, as the mathematical notion does. This also illustrates
that something resembling the mathematical notion _is_ available to us
in everyday life; but it is not clearly marked off from the original
notion.
Of course, the committee organizer does not have the problem we
outline above, even if he is entirely innocent of set theory. He
makes a list of possible subcommittee rosters and easily manages to
count the four 3 member committees and the 2 that do not contain both
A and B. Notice that he/she manages this (without being conscious of
having done anything special) by representing the subcommittees (which
themselves overlap) by non-overlapping configurations of marks on
paper which can be counted in a sensible way.
Consider the sets {a,b}, {b,c}, and {a,c}. If collecting things
together into a set was simply a kind of fusion (terminology due to
Lewis), there would be no way to tell the difference between
{{a,b},{b,c},{a,c}} and {{a,b},{a,c}}. The sets {a,b}, {a,c}, and
{b,c} can be "fused" into the set {a,b,c} which is the smallest set
which has all of them as "parts" (subsets), but the "fusion" of {a,b}
and {a,c} gives the same result. The actual move we take to construct
distinct sets {{a,b},{b,c},{a,c}} and {{a,b},{a,c}} involves first
getting the disjoint (and thus non-overlapping in terms of our
understanding of part and whole for sets) objects {{a,b}}, {{a,c}},
and {{b,c}}. All subcollections of a collection of pairwise
non-overlapping objects have distinct "fusions", and we get distinct
sets by "fusing" the singletons corresponding to the elements of each
set.
There isn't any natural objective correlate to the singleton
construction as there is to the part-whole relation (this is why Lewis
professes to be mystified by it). But it has an obvious function in both
the concrete example and the abstract example above. The problem
which needs to be solved in both cases is that we want to be able to
distinguish between collections of overlapping objects which have the
same "fusion"; the solution is to choose objects from a domain of
non-overlapping objects to _represent_ the overlapping objects. In
the everyday example, the role of representation is transparent; the
committee organizer is probably not even conscious of a problem. My
suggestion is that the role of the singleton construction is
_semantic_; a singleton is a kind of _name_ for its element; a way of
putting it which is perhaps better is that the singleton is a _token_
used to replace its element in the construction of sets.
The final suggested formulation of the notion of mathematical set is
as follows: for certain objects ("elements") we provide a collection
of pairwise non-overlapping "names" or "tokens". The set with certain
elements is the fusion of the tokens corresponding to those elements.
This notion, unlike the intuitive notion of collection, admits
iteration: some fusions of tokens ("sets") may themselves have tokens
assigned to them, and so be elements in their turn. Notice that this
notion does correspond to an everyday idea: a fusion of names of
objects is a _list_ (abstracting from the order of the items).
This formulation of the notion of "set" avoids (and perhaps illuminates)
Russell's paradox:
Let R be the "set of all sets which are not elements of themselves".
A "set" in our interpretation is "a fusion of tokens". A "set" is
"not an element of itself" if its token is not part of the fusion it
stands for.
So R is the fusion of all tokens x which stand for fusions y of tokens
such that x is not part of y.
If there is a token r for R, is it part of R? If r is part of R, then
r should not be part of R, by definition of R; if r is not part of R,
then it should be part of R.
The conclusion we draw is that there is no token r. Notice that we do
not conclude that there is no fusion R; I think that it is reasonable
to assume that any collection of objects whatever has a fusion (and
that this natural assumption provides misleading intuitive support to
the naive axiom of comprehension). But there is no convincing
intuitive principle which would force us to believe that any fusion of
tokens whatsoever can itself be assigned a token.
Once we observe that not all fusions of tokens can have tokens assigned
to them, we introduce the following definitions:
"class" -- fusion of tokens
"set" -- fusion of tokens represented by a token
"proper class" -- fusion of tokens not represented by a token.
All of this is an exercise in ph.o.m. rather than f.o.m.; there are no
mathematical consequences to be drawn, per se. But the crucial role
which the notion of "set" has in f.o.m. makes its intuitive
underpinnings fair game here.
This analysis illustrates that the mathematical notion of set is not
an obvious intuitive notion. At best, it is abstracted from a
secondary strand in the usual notion of "collection"; analogies with
the dominant everyday notion of "collection" can be seriously
misleading (this happens in practice with students when they first
encounter these ideas). This analysis only becomes relevant when sets
of sets are considered; if one is considering sets of real numbers (as
Cantor was in his original work) the usual notion of collection is
adequate, because the real numbers can be thought of as "points" (and
confused with their singletons!), as long as they are not themselves
being considered as sets.
This analysis suggests a "second-order" view in which proper classes
are treated as first-class objects (if one took this view of ZFC,
Morse-Kelley set theory would recommend itself). The existence of
fusions of arbitrary collections of objects seems to be a reasonable
assumption, and so "classes" in this picture seem just as real as
"sets" (and so quantification over proper classes seems legitimate).
Finally, this analysis makes the construction of the iterative
hierarchy seem more problematic, in what I think is a salutary way.
There is nothing in this picture to suggest that ZFC is inconsistent.
What it does make clear is that one cannot expect to get the new sets
at each level "for free": when one has constructed all elements of
V_{\alpha} and provided them with tokens, all elements of V_{\alpha+1}
are indeed "given for free" (as arbitrary fusions of tokens
representing elements of V_{\alpha}), but one needs to make a further
trip to the supply of tokens (wherever it is) to get new tokens to
represent each element of V_{\alpha+1} in order to proceed further.
The difficulty is the one which Russell found in his axiomatization of
the theory of types: it is necessary to assume that there are "enough
objects". The axiom on which this casts some doubt is Power Set.
None of this is to say that I don't advocate set-theoretical
foundations and rely on the intuition of the iterative hierarchy of
sets. I in fact do advocate set-theoretical foundations (and some
equivalent alternatives, mainly formulations in terms of functions)
and I am convinced of the validity of the intuition of the iterative
hierarchy (and am willing to admit the reasonableness of some stopping
points short of full ZFC, as well as some beyond ZFC).
Comments invited...
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.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes
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