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Topic: The finite infinite and the infinite infinite
Replies: 48   Last Post: Sep 4, 2012 4:00 PM

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 David Hartley Posts: 463 Registered: 12/13/04
Re: The finite infinite and the infinite infinite
Posted: Aug 24, 2012 6:40 PM
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In message
<cc4504cf-e8b4-4a94-a1a0-d9c1785ce5f5@c4g2000yqi.googlegroups.com>,
MoeBlee <modemobe@gmail.com> writes
>On Aug 24, 3:02 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>> If there were sets that
>> could not be well-ordered, then they had no ordinal number. And then
>> their cadinal number could not be fixed.

>
>That is, if we use a certain definition of "card", yes.

There is no associated aleph, yes, but the set still has cardinality and
a cardinal can be defined using "Scott's trick".

>> There would exist sets A and
>> B for which the cardinalities were not in trichotomy.

>
>More exactly, it would be the case that we can't prove that for every
>set A and B, either card(A) =< B or card(B) <= card(A).
>
>It would NOT be the case that we could provide examples of sets A and B
>that don't satisfy card.
>
>Most importantly, it would NOT be the case that we could prove:
>
>There exist sets A and B such that A and B fail trichotomy.

Are you sure? If A cannot be well-ordered but B can, then
card(B)<= card(A) is possible, but card(A) <= card(B) is not.
However, you can not have card(B) <= card (A) for every well-orderable
B, so there will be some B which is incomparable with A. In particular
there will be a least aleph incomparable to A. It is called the Hartog's
number of A. http://en.wikipedia.org/wiki/Hartogs_number
--
David Hartley

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