Re: Review of Mueckenheims book.

REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 3 of the Reply)


In number ten the author attempts to show that the first proof by
Cantor
of the uncountability is invalid. That proof is given on page 77
of the
book, but let me recap. Given a list of real numbers, it is
possible
to find for any arbitrary interval (a, b) a number in that
interval that
is not in the list. The proof is by a sequence of nested
intervals in
(a, b), where the first is found by the first two numbers in the
list
that are larger than a and smaller than b. We replace a by the
lower
number and b by the larger number and repeat the process, Going
further
through the list. This way a list of nested intervals is found
that is
either finite, or is infinite. If infinite, the limits of the
lower
bounds and the upper bounds can be equal or not. If equal, that
limit
is the required number, if not equal, each number in the interval
given
by the limits can be used. If the list of intervals is finite, we
take
the last one. That one can contain at most one more number from
the
list, so we can take every number from that interval, except the
additional number, if it is in there. In this section it is clear
that the author does not understand the basics about how that
shows
that the reals are not countable.

WM:
How could this simple stuff be not understandable? Of course it is
easy ti understand, but it seems not so easy to understand my critics,

Because the proof above is valid
for *every* list of reals, a list of reals does not exist (every
list
is doomed to be incomplete), and so the reals are not countable.

The proof must be valid for every list of reals. Yes. Therefore it
must also work for a very artificial list.

Now the author tries to use the proof to show that even countable
sets are uncountable. He starts with the sequence of rationals:
-1, 1/2, -1/3, 1/4, ...
and concludes from that that the (single) limit is 0, not in the
set
used, and attempts to show by this that the set of numbers in the
sequence with 0 added is uncountable. The author fails to see one
important difference: in the set of reals, each open interval
contains
reals, in the set of rationals from the sequence above that is not
true. But that *is* necessary to make it work.

Each open interval of rational contains rationals. They can be treated
like the sequence above by a simplke coordinate transformation.
But what I said in the book is that from th result of Cantor's first
proof alone one cannot distinguish between the cardinal numbers of
transcendental and rational numbers.

In eleven the author attempts to do the same with the second
proof.
(Although the author disagrees with me, the second proof is
actually
not about the reals, but about sequences of binary digits without
interpretation, somebody has later modified it to a proof about
the
reals. Was that Zermelo (see below under twenty)? In the
original
version it proves the theorem that there are uncountable sets.)
But
in this case it is in the modified, decimal, version (who came
first
with the decimal version?). Here he does not even show a
contradiction,
or whatever, but only asks how it is possible that a complete list
indeed *does* exist, and asks whether that is the right way to
look at
the infinite. But that is not a mathematical but a philosophical
question. Here he shows only a contradiction with his
sensibilities
(remember Bolzano's musings in chapter 6?).

In twelve he attempts to find a contradiction, and builds a list
(ternary I think),

Doesn't matter. The idea was decimal, but the digits 3 to 9 are not
used.

where in the building of the diagonal every 0 and 1
is replaced by a 2, and concludes that the diagonal found at every
step
is in the list. Strange that he sees that as a contradiction,
because
in the building of the diagonal *each* digit has to be changed.

But it cannot be done other than step by step.

I hesitate to name the next one thirteen, because it is strongly
connected with the previous, but the authors now gets some quite
different conclusions. This example is also repeatedly mentioned
on
sci.math, it is about the list 0.000..., 0.1000..., 0.11000...,
0.11000..., etc. The 0 in the diagonal is replaced by a 1 and he
shows
that the diagonal as built up to the n-th row is equal to the (n
+1)-th
row. As he rightly states, the limit (when seen as decimal: 1/9)
is not
in the list. The proper conclusion from this is that the list is
incomplete. However, the author comes to a quite different
conclusion,
for this purpose he constructs a new list, where the first 0
(everything
is happening after the decimal point) is replaced by 1. He states
that
the diagonal of that new list does not exist because is should
have
aleph-0 1's, and that would require that there is a list element
with
aleph-0 1's. He tries to show that with finite segments. But
this
fails, obviously, as the (complete) diagonal is constructed with
limits
in mind.

There are limits in set theory?
Using the tools of set theory we can define a bijection. An existing
diagonal with X elements guarantees the existence of as many lines and
columns.

Fourteen is also closely connected. Obviously the author is
thinking
that the creation of the diagonal is an iterative process, which
is
*not* the case.

Which *is* the case. You cannot start at the end because it is not
there. So you cannot start with anaction including the end. So you can
only start at the first or at least a finite position n.


Given a list, you can refer to the n-th digit of the
created diagonal without ever referring to any other digit of the
created diagonal.

Yes, but with the n-th digit you have not all of them. And you cannot
start at he non existing end. So you must start at a finite oposition.

The creation of the digits are independent of each
other. This is probably because the author has not read much
beyond
Cantor, who indeed saw everything as sequential processes.

Of course the natural numbers are a sequence. In Cantor's case, the
sequential construction leads to contradictions. Some advocates of set
theory have recognized this and, therefore, postulated that everything
happens at once, but that does not mean that this position is correct
or sensible. One cannot do infinitely many transactions at once.
So the
author is thinking that if a sequential process (creating the
diagonal)
can be done in one step, so can every sequential process. A deep
misunderstanding.

LOL.

If operations do not depend on each other, you can
do them all at once, if they do depend on each other you can not.

And the existence of the number digit n+1 depends on the existence of
the digit number n.

Regards, WM

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