Re: ** says: Definition: sum{i in N} i = 0

On 8 Jul., 22:50, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 8 Jul., 20:38, hagman <goo...@xxxxxxxxxxxxx> wrote:



On the contrary it is simple because Cantor himself considered his
proof an existence proof for trancendental numbers.

Well, but that does not make transcendental number the *cause* for
the uncountability of the reals.
For example, the existence of transcendentals is also possible
by approximation theory: sum 10^(-n!) is transcendental.
Thus transcendental numbers exist. But this tiny fact doesn't
prove uncountabilty of the reals.

Of course only the (asserted) existence of *all* transcendentals makes
R uncountable. Those few transcendentals which were constructed by
Liouville himself and the handful being found later on by Hermite, v.
Lindemann, Schneider, Gelfand and others is certainly not sufficient
to make anything uncountable. (By the way, it way Liouville's aim
already to prove e transcendental.)

Let a_0 be an sequence of ones and twos.
Then
sum a_n*10^(-n!)
is transcendental of Liouville type and the set of these is
uncountable,
of course by the same diagonal argument as used for the reals
themselves.


Anyway, saying that A has cardinality c is "caused" by the existence
of subset B with cardinality c sounded somewhat strange to me.

Remove the transcendentals from R and investigate the cardinality of
the remaining set. Dedekind already showed that the algebraic numbers
are countable. A point which my students learned in lesson 11.

One might ask what causes the latter etc. an d will not only never
arrive at a "causa prima" but not even ever at a good point to
stop the search as anything one can come up with is either of
cardinality c as well or of cardinality aleph_0 and thus too small
for being a "cause" for uncountability.
With an axiom "There is a subset X of the reals that has
cardinality strictly between alpeph_0 and c" viewing such a set
as a "cause" would seem more plausible to me, but the notion
of causality in math is to be taken with care.
It's as if the reals had been countable in former times but were
forced to change their cardinality when the transcendentals "came
into existence".

That is correct.



Man, I must admit that out of the belly I'm quite badly prepared
for the exact historical facts asked for in these questions;
that's really something where I usually *must* rely on handwaving.

You would have had plenty of time to search the net or printed
literature.

Which I did not as that would have been unfair (or did your students
have comparable online research tools available duing their test?)
OTOH, they had a chance to prepare based on your material.

I pointed them to Wikipedia (english) and other sources, weeks before
the test.

Regards, WM


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