Das Kalenderblatt 091025

24/10/2009 - 15:01 von WM | Report spam
{{P.E.B. Jourdain hatte soeben, in KB 091024, eine absolut
(=potentiell) unendliche wohlgeordnete Menge U aller Ordinalzahlen
definiert:}}

Also, we may define aggregates (W, m_1) {{es scheint, dass die
Mengenlehre mit der Kombination WM hàufig in Konflikt geràt}}, where
the element m_1 follows all the elements of W, and so on; we must, in
fact, say that W is similar to a segment alone of a (well-ordered)
series U such that every well- ordered series is similar either to it
or to a Segment of it. The conception of U excludes the contradiction
that suggests itself if we define an element subsequent to every
element of U, for if we could so act, our U could not be the U first
defined; in words, U is an absolutely infinite series.

Now, it is quite evident that the elements of any aggregate (M) can be
arranged in a series similar to U or to a segment of U. For if we
conceive any elements to be removed successively from M, beginning
with the series
(3) m_1, m_2, ..., m_nü, ..., m_omega, m_omega+1, ..., m_gamma, ...
we ultimately exhaust the given aggregate M; for if we did not so
exhaust it, there would be at least one element (m') remaining {{jaja,
an dieser Stelle sei es wiederholt: so denken die Leute, die überall
binàre Bàume pflanzen: Es geht immer so weiter, denn wenn nicht, so
gàbe es ein erstes m' à la Jourdain}} and, accordingly, we could form
a well-ordered aggregate of which U was a segment.
This last argument, now, seems to me to be the essential part of
Zermelo's proof; for the 'gamma-covering' used as a basis for the well-
ordering of the elements of M is not necessary, and, I think, obscures
the point at issue. {{Das ist der Sinn der Sache. Sonst merkte eh
jeder gleich, dass nach Zermelo (s. KB091009, KB091010) jede Menge
linear und ohne Unterbrechung wohlgeordnet werden kann - im
Widerspruch zu Cantors Satz.}} [...]
But, inversely, the series (3) is easily seen to define a gamma-
covering, provided that the series (3) ultimately exhausts the
elements of M, as we have seen that it must do.
It is not, then, necessary to use the artifice of a gamma-covering. We
can more simply imagine the series (3) built up without this, and the
essential part of the proof is the same in both cases.
[P.E.B. Jourdain: "On a Proof that every Aggregate can be well-
ordered" Math. Annalen 60 (1905)]

Gruß, WM
 

Lesen sie die antworten

#1 Robert Figura
24/10/2009 - 19:23 | Warnen spam
Hallo WM

Ich kann ja damit leben daß Du Deine Ideen, obwohl hier Themenfremd,
tàglich Postest um Dir und Anderen die Illusion zu bereiten es seien
relevante Ideen. Daß Du aber inzwischen darauf zurückgreifen mußt
denselben Text mehrfach zu Posten kann ich nicht gut finden.

Denk doch bitte noch einmal darüber nach ob Du es Dir mit Deinem Talent
nicht doch leisten kannst eine gepflegte Konversationskultur aufrecht
zu halten.

Mit freundlichen Grüßen
- Robert Figura

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