FOM: On one feature of Cantor's Diagonal Method
Alexander Zenkin
alexzen at com2com.ru
Sun Oct 18 21:37:53 EDT 1998
Dear [HM]-friends,
I have one "historical" question about the most important FOM-object -
on the Cantor Diagonal Method (CDM) - of the following kind.
As is well known, the essence of the CDM is as follows.
Let
x1, x2, x3, ..., xn, ... , (1)
be an infinite sequence of real numbers, say, of the segment [0,1].
For simplicity, we shall use the binary number system.
Applying the CDM to the infinite diagonal,
0, x11 x22 x33 ... xnn ..., (2)
of the sequence (1), we define (construct, produce, generate and so on)
a new real number, say,
y = 0, y1 y2 y3 ... yn ..., (3)
according to the DCM-rule: for any i>=1,
(if (xii=0) then (yi:=1)) and (if (xii=1) then (yi:=0)), (4)
It is obvious that the Diagonal (more precisely, anti-Diagonal) number
y differs from every (and,
therefore, from all) numbers of the sequence (1).
But why do we always speak about the diagonal (2) of the given
enumeration (1)?
I can take any real number of the segfment [0,1], even, for simplicity,
the real number "identical zero":
0, 0 0 0 ..., (5)
and, using the same enumeration (1) and the same CDM-rule (4), and
repeating, word for word,
said above, I define (construct, produce, generate and so on) the same
anti-Diagonal real number (3).
So, my question is such: whether anybody and anywhere used such approach
for the proof of Cantor's Theorem on uncountability of the set of all
real numbers of the segment [0,1]?
P.S. Such the approach allows to apply the DCM to any finite
enumarations of real numbers, say,
x1, x2, x3, ..., xn, (1a)
and, at any step of such the application (and, of course, after its last
n-th step), we shall get not only real, but even rational (with the
0-tail) anti-Diagonal number:
y=0, y1 y2 y3 ... yn 0 0 0 ...
But this means that the DCM does not distinguish finite sets from
infinite sets, i.e. the DCM does not distinguish sets by their
cardinality :-)
Some other related rather surprising results are discribed in my
recent paper: Zenkin A.A., "The Time-Sharing Principle and Analysis of
One Class of Quasi-Finite Reliable Reasonings (with G.Cantor's Theorem
on the Uncountability as an Example)". - Doklady Mathematics, vol 56,
No.
2, pp. 763-765 (1997). Translated from Doklady Alademii Nauk (Section
"Mathematics"), Vol. 356, No. 6, pp. 733 - 735.(1997).
The English e-copy (MS Word for Windows'95) of the paper is available in
the Section "FOUNDATION OF MATHEMATICS" at my WEB-Homepage
http://www.com2com.ru/alexzen
Best regards,
A-Z
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