FOM: Transfinite Logic

[wiman lucas raymond]

> was entirely correct, but they were using a faulty definition (if what
they
> quoted was correct, and not taken out of context).

The definition is faulty in its abiguity.  What the author meant (I'm
sure), is that a subset A of a set B, A is proper if there exists an x
in B such that x is not in A.  Unfortunately, the way the writer phrased
it, it can be confusing to students.  It's bad pedagogy at the very
least, incorrect mathematics at the worst.

>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.   

Quite right!  It is precisely because of this requirement for intuition
that these proofs are often quite fallacious.  Let us assume the axiom
of choice (and hence the well-ordering principle), and let < be a
well-order on the reals.  Then it is quite possible to "pair up" the
reals with the natural numbers "proving" that the reals are countable:

1     2     3     4   ...
x_0 < x_1 < x_2 < x_3 ...

Here x_n is the n-th smallest real number under the well-ordering <.  Of
course what's fallacious is that there is also an x_omega (where omega
is the smallest infinite ordinal), and an x_lambda for every ordinal
lambda up to omega^omega (but no natural numbers with which to pair
them).  Still, since we never see these bigger numbers, with this kind
of an intuitive argument one can "prove" that every infinite set is
countable.

However, we need not use the pedagogical crutch of an intuitive
argument, as we can construct the required bijection.  The set of all
"tens" is the set of positive integers divisible by 10, explicitly T =
{x in Z^+: there exists c in Z such that x=10*c} (Z^+ is the positive
integers).  Then the function f: Z^+ -> T defined by f(x)=10*x is one of
the possible bijections (the proof is very easy).

>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.

What is the difference between "matching" and having a coorespondence?
How does the above bijection not match the elements of the two sets
"without remainder"?

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

Perhaps you see this argument as "tired" and "old", but calling it names
doesn't prove anything about it.  Somehow, I don't find the few
paragraphs you used to make your point convincing.  Why is this
tired, old argument wrong?

- Lucas Wiman