[FOM] Plural quantification and set theory

Dean Buckner Dean.Buckner at btopenworld.com
Sat Sep 13 11:11:48 EDT 2003

William Tait writes ("Non-distributive predication" 14 July), concerning Tom
Mckay's system of plural quantification

> there is a more or less obvious translation of second-order
> predicate logic (where the second order variables range over
subsets of the domain D of individuals) into [McKay's system]
> and conversely---the pluralities corresponding to non-empty
> sets.

>But for this reason, how can [McKay's] theory have any foundational
>significance? One understands your notion of plurality over a
domain D exactly to the extent that we understand the notion of a
> non-empty subset of D.  And any contradiction arising for the one
> notion immediately translates into a contradiction for the other.

>So why not, when we are just dealing with a domain and the
> subsets/pluralities over it, simply opt for the former, since it is
> most familiar and would involve no revision of our customary
> mathematical teminology?

I agree with this, if there such a translation, but I have a further point.
As well as there being no correlates in plural theory of empty set and
singleton set, I'm not sure there is any correlate of an infinite set.
Consider the function f such that

    f(1) = the numbers 0 & 1
    f(2) = the numbers 0, 1 & 2
    f(3) = the numbers 0, 1, 2 & 3
    f(n) = the (consecutive) numbers 0 - n

The function maps natural numbers onto the plural entities defined in
McKay's system.  Now consider

            N = { n: (E s) s = f(n) }

N is the set containing every finite number, and nothing else, since to
every finite number n there corresponds some consecutive sequence of numbers
from 0 to n.  But the existence of N is consistent with there being no
"infinite pluralities", i.e. without there being anything but what is
referred to by "the numbers from 0 to
n" for finite n, and so nothing referred to by "all the numbers in N".

This results in the apparent paradox that the set N contains every natural
number, but there are no things corresponding to "the natural numbers".  But
it's only apparent: it's paradoxical only for those who insist on reading

    {x: x is F}

as "the things that are F" or something like that.  "The natural numbers" is
not an expression that belongs to set theory.  It's an English definite
description, and all such descriptions refer to the things that satisfy the
description, unlike the corresponding sets.

This underscores the point made by Martin Davis, and also William Tait (in
the same posting further down) that set theory represents a language in its
own right.  We run into paradox only when we try to interpret or "translate"
set theory in ordinary language.

The same probably applies to much of the endless philosophical debate around
set theory, for example about "completed" infinities.  The term "completed"
has as far as I know no meaning in set theory itself.  The nearest we get is
the idea that there is a set N that contains every finite number there is,
and that is beyond dispute, if we accept the standard axioms of the theory.
In the sense that there are objects that we can refer to as "the" natural
numbers, that is doubtful.

It is not helped by tendency to conflate "the set of F's" with "the F's".
In ordinary language, if I wash a set of dishes, I just wash the dishes.
but I do not wash

    {x: x is a dish}

do I?

This point, that ordinary talk about more than one thing is not to be
confused with set-theoretical talk, is not to be confused with the question
raised by Martin Davis some time ago, whether we can dispense with set
theory and do mathematics in ordinary language, or some formal system that
more closely resembles ordinary language.  That question is still open.  I'm
not ignoring it, I just don't know the answer.  I wish I did.

"What set theory has to lose is rather the atmosphere of clouds of thought
surrounding the bare calculus, the suggestion of an underlying  imaginary
symbolism, a symbolism which isn't employed in its calculus, the apparent
description of which is really nonsense."

Dean Buckner

Work 020 7676 1750
Home 020 8788 4273

More information about the FOM mailing list