Re: Cantor and the binary tree

In article <MPG.1d2dcd001d42d0ca989f19@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Matt Gutting said:
> > Tony Orlow (aeo6) wrote:
> > > Matt Gutting said:
> >
> > <snip>
> >
> > >>David is right in saying that the proof does not depend on a
> > >>diagonal; it can be rephrased without even relying on any
> > >>geometric analogy, and still holds true.
> > >
> > > Give it a shot. I'd love to hear that form of convoluted
> > > nonsense. I have never heard any other real version of this
> > > "proof". Enlighten me.
> >
> > Okay, here goes.
> >
> > Let the sequence {a_i}, indexed by the natural numbers, be a list
> > of all real numbers in the closed interval [0,1] (or, if you like,
> > [0,.9999999...]). That is, suppose we can make an ordered list of
> > all the reals in this interval, and suppose further that we can
> > label each real a_1, a_2, a_3, etc.
> >
> > Now each real a_i is equal to the limit, as n increases without
> > bound, of sum(j goes from 1 to n)[d_ij * 10^(-j)], where d_ij, the
> > jth (let's say decimal) digit of a_i, is a nonnegative integer less
> > than ten.
> >
> > (It is important to note that a_i is not somehow "constructed" by
> > taking partial sums; it is _defined_ as a single number that is a
> > limit of these partial sums. It is possible to evaluate a_i as the
> > limit without evaluating any of the partial sums.)
> >
> > Now, define a sequence, or rather an ordered list, of nonnegative
> > integers {e_j}, such that e_j = 1 when d_jj = 0, and e_i = 0 when
> > d_jj <> 0. It is clear that such a sequence exists; further, just
> > like a_i, it is not "constructed" by evaluating each e_i in turn,
> > but exists all at once, according to the definition. It is not even
> > necessary to determine the values of any member e_i in the
> > sequence.
> >
> > There exists a real number e which is the limit, as n increases
> > without bound, of sum(j goes from 1 to n)[e_j * 10^(-j)].
> >
> > Now any real number in the interval can be represented as I
> > represented a_i above, or as I represented e. Further, any two
> > numbers expressed as
> >
> > limit as n increases without bound (sum (j from 1 to n)[s_j *
> > 10^(-j)])
> >
> > and
> >
> > limit as n increases without bound (sum (j from 1 to n)[t_j *
> > 10^(-j)])
> >
> > are equal if and only if s_j = t_j for all j (that is, the two
> > numbers are equal if and only if all corresponding digits are
> > identical).
> >
> > But for all j, e_j <> d_jj; thus e differs from each a_j in at
> > least one place (the jth place); thus e is not equal to any a_j in
> > the list, so that e is a real number in the given interval but not
> > in the list. But by hypothesis, the list {a_j} included all the
> > real numbers in this interval, so that e *must* be in the list.
> > This is a contradiction; therefore our hypothesis (that I can have
> > a list of all reals in this interval, and index this list with
> > natural numbers) must be false. Therefore, there are more reals in
> > this interval than can be indexed with the natural numbers; that
> > is, there are more than countably many reals in the interval.
> >
> Indeed it shows that there are more numbers than digits, so there are
> more reals this way than naturals, it would seem, but it doesn't
> prove that e is not anywhere on the list. e is a number whose index
> is greater than N, but which is on the list. That is, e is a number
> for which there is no d_jj, because N elements have already been
> listed. The bogus assumption here is your hypothesis that if this
> contradiction arises, it is because the list is not enumerable, when
> really the reason is that it always contains more numbers than
> digits. This seems obvious to me, and the conclusion normally drawn
> seems to be based on an unfounded assumtpion that all countable
> infinities are the same, based on bijections with sloppiness, and
> this mediochre use of counting to define infinity.

The basic definition of a *list* of members of a set S is a mapping
f: N --> S.

So that, contrary to TO's delusions above, anything not indexed by a
member of N CANNOT be in the list.

What seems so obvious to TO involves ignoring and contradicting
essential properties of what he is talking about
.

Relevant Pages

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