The Second Main Theorem of MatheRealism

24/05/2014 - 09:13 von Jürgen R. | Report spam
Theorem 2 (Mückenheim 2004):
Let {I_n} be an infinite but countable set of disjoint intervals in (0,1),
ordered
by magnitude, and let {a_n} be any dense sequence of different points
in (0,1). Between {a_n} and {I_n} a one-to-one correspondence can
be established, such that the relative position of a_{k_n} and a_{k_m}
is the same as that of I_n and I_m.

(Added hypothesis: Each I_n has non-empty interior.)

Proof: Since the a_n are dense in (0,1), for each I_k there is an a_{n_k} in
I_k,
and these are necessarily ordered as required.
The proof is continued by showing that this is really a one-to-one
correspondence.

Note 1:
By "the proof is continued" Mückenheim means a rather famous proof of
Cantor's
(page 236-244, Ges. Abhandlungen), where it is shown that every nowhere
dense perfect set in [0,1] is uncountable. The argument, of which
Mückenheim
has extracted a part, shows that the intervals of the complement of any
nowhere dense perfect set have the same order structure as a countable
dense point set in [0,1].
The trouble is that Mückenheim forgets the essential hypothesis. Cantor
doesn't prove the assertion for arbitrary intervals, but only for a
very particular collection of intervals. So this theorem is
valid in MatheRealism, but obviously not in Matheology.

Note 2: I have stated the "theorem" verbatim, but have reduced the "proof"
to its essence.

Ref.
A severe inconsistency of transfinite set theory
W. Mückenheim
University of Applied Sciences, D-86161 Augsburg, Germany
[mueckenh@rz.fh-augsburg.de]

Talk delivered at the meeting of the German Math. Soc., section logic, in
Heidelberg, Sept. 14, 2004.
(Believe it or not - apparently true - but not published in the Proceedings
of the meeting )
 

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#1 Jürgen R.
24/05/2014 - 09:16 | Warnen spam
Entschuldigung - war nicht für diese mückenfreie Gruppe gedacht

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