Kardinalzahlvergleich und Wohlordnung.

08/02/2016 - 19:17 von WM | Report spam
What remains is to show that
given two sets A and B, one is less than or equal to the other.
If one thinks of this problem for two "arbitrary" sets, one sees the
hopelessness of trying to actually define a map from one into the other.
I believe that almost anyone would have a feeling of unease about
this problem;
{{außer einigen Experten in dsm}}
namely that, since nothing is given about the sets, it
is impossible to begin to define a specific mapping. This intuition is,
of course, what lies behind the fact that it is unprovable in the usual
Zermelo-Frankel set theory. Cantor suggested a method of proving it.
It depended on the notion of a well-ordering, i.e., an ordering of a set
A in which every nonempty subset has a least element. If A and B
have well-orderings, it is not hard to show that either there is a unique
order preserving map of A onto an initial segment of B, or vice versa.
So if A and B are well-ordered, we can define a unique map which
shows that the cardinality of A is less than or equal to that of B,
or conversely. By an ordinal we simply mean an equivalence class of
well-orderings. It follows that the ordinals are themselves well-ordered.
Now if one assumes the well-ordering principle, that all sets have a
well-ordering, it follows that all the cardinal numbers have at least
one ordinal of that cardinality. Thus, we have in the "sequence" of
all ordinals, particular ordinals which we now call cardinals, which are
defined as ordinals whose cardinality is greater than that of any of its
predecessors.
[Paul J. Cohen: "The discovery of forcing", Rocky Mountain Journal of Mathematics 32, 4 (2002)]
 

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#1 Jürgen R.
08/02/2016 - 20:03 | Warnen spam
On 08.02.2016 19:17, WM wrote:
What remains is to show that
given two sets A and B, one is less than or equal to the other.



Nobody in heaven or on earth can prove that
"given two sets A and B, one is less than or equal to the other".

The reason is that the statement is nonsense in the absence of a
specific order relation.

You are quoting out of context from an informal talk. The speaker
doesn't mean what you are insinuating.

Evidently the quote, even after you have suppressed the context, doesn't
serve your purpose. Cohen here is informally discussing the
Cantor-Bernstein theorem. This has nothing to do with your mistaken
claim that bi-unique maps depend upon well-ordering.

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