FOM: Transfinite Logic

[Dean Buckner]

In a case reported in a discussion group, a group of students failed a
question on set theory that, they claimed, had been a textbook answer.  The
book _Fundamentals of Mathematics_, by William M. Setek) is widely used as a
college textbook on mathematics and set theory.  The question was whether
the set A = {10,20,30 ...}was a "proper subset" of the set N = {1,2,3 ...}.
The students thought it not, since the definition they used was:

    A proper subset contains at least one less element than the parent set.

which they claim was in Setek's.book (should be easy to check).  They argued
that, since N is clearly equinumerous with A, A cannot contain "one less
element".  Their professor failed this answer since, according to any
standard definition, {10,20,20 ...} is a proper subset of the natural number
"set".

This raises some difficult  questions.  To my mind the students' reasoning
was entirely correct, but they were using a faulty definition (if what they
quoted was correct, and not taken out of context).  The definition should
be, i think

(a) A proper subset is such that the parent set contains at least one
element which is different from every element in the subset.

or (assuming an appropriate criterion of identity)

(b) such that the parent set is not identical with the subset

It also raises the interesting question of how transfinite logic subverts
our ordinary intuitions about number and collections, and how writers of
textbooks can sometimes be victims of this.

It raises the final question, which those who have followed my other
postings will recognise, as to whether a proper subset (as defined) can be
equinumerous with its parent, given my definition of "less than".

I have argued that the concept of number is closely bound up with that of
identity and difference.  I argued ("Barbie hath said") for the following
equivalences

    there is one thing =df there is 1 thing
    there is one thing and another thing =df there are 2 things
    there is one thing and another thing and another thing =df there are 3
things
    ....

I also argued ("Numbers of Objects") that "The predicate "x is a different
thing from y" is a 2-place predicate, satisified by all couples of things
(Caesar and Anthony, Anthony and Cleopatra, Caesar and Cleopatra ...).  Its
terms are singular ("Cleopatra", "Caesar" ...) and thus signify individual
things, not concepts."

No one disagreed with this - many thought it was trivial.  But it's not!
Given that a parent set has at least one member x that is numerically
different from every member of any of its proper subsets, it follows that x
is "another" member.  Assuming we can move from "another member" to "one
more member", and thence to "a set that has one more member than another
set, cannot have the same number of members as the other set", it follows
that no proper set can be equinumerous with its parent.

Before the usual hasty replies, I have no quarrel with the standard
arguments about this.  My quarrel is foundational, about how we define
notions like "identical", "subset" "equal-numbered" and so on.  I will say
this, though.  the usual "proofs" that start off with

    1   2   3   4 ...
    10 20 30 40 ...

rely on some sort of intuition as to what the dots mean, and what
correspondence is.  We have to spot that "every" member of the second series
is a member of the first (is that so?).  We have to spot that every member
of the first "matches" every member of the second.  We then have to move
from the idea of correspondence to that of equal number.  But this relies on
the very same instincts that underlie my definition of number: that
equinumerosity is matching "without remainder", that if there was just
remainder, there would be "one more" left over, and mere correspondence
would not imply equal numbers.

In summary, the standard argument that (e.g.) the set of tens has the same
number as its parent set of natural numbers is based on a deep-rooted
intuition about what sets are, that the argument is at war with.  As Andrew
Boucher says, in a Johnsonian piece I rather like, surely the natural
numbers should have "very many more" members than the set of tens, evens or
other?

I will expect the usual tired old argument "what you say is valid only for
finite sets".



Dean Buckner
4 Spencer Walk
London, SW15 1PL
ENGLAND

Work 020 7676 1750
Home 020 8788 4273