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Das Kalenderblatt 091025

{{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 häufig in Konflikt gerät}}, 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
binäre Bäume pflanzen: Es geht immer so weiter, denn wenn nicht, so
gäbe 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

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