Das Kalenderblatt 100703

02/07/2010 - 12:01 von WM | Report spam
Cantor’s Problem {{Ende}}

It can therefore be concluded that there exists no set Q having
cardinality greater than the cardinality of S, as any set Q can be
shown to be either in Q1 or Q2 as defined above, and thus “contained”
by correspondence with a set contained in S through an injection into
S. The later is a result of considered importance to the conclusions
that Cantor’s Problem does admit.
Now to the essential question of whether R has an injection into any
set contained in S. Since R is represented by strings over the set P’
= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, ‐, .}, a computationally-countable
(finite) set, it can be concluded that S contains at least one set
into which an injection from R exists because R is in Q1 as later was
defined above. Such a conclusion contradicts Cantor's conclusion as to
the relative cardinality of R and Z; given that it has been shown that
Z has a bijection to S by constructive proof and R has at least an
injection into S. If we assume that R exists in a bijection relation
to S, such grants at best |R| = |Z|. [...] the set C = {c | c {m, w}
* } [...] is contained in Q1, as defined above. By direct substitution
of symbols m-->0 and w-->1, any string in C can be rewritten to a
string in S. Given that there does not exist in C two strings c1, c2
such that c 1 = c2, the rewriting of strings in C produces by
definition a bijection with S. This results in a contradiction for two
reasons. The bijection with S is such that C is denumerable by the
same means with which S is placed into bijection relation with Z.
While at the same time, S is not denumerable if the Diagonalization
argument is valid. The latter then stands in contradiction of the
bijection between Z and S.
[Charles Sauerbier (2009)]

Gruß, WM

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#1 Rainer Willis
02/07/2010 - 13:59 | Warnen spam
Am 02.07.2010 12:01, schrieb WM:
Cantor’s Problem {{Ende}}


[Charles Sauerbier (2009)]

"The file is damaged and could not be repaired."
Kein Wunder.

Gruß Rainer

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