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Topic: Uncountable List
Replies: 8   Last Post: Dec 31, 2012 3:34 AM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Uncountable List
Posted: Dec 31, 2012 1:55 AM
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On Dec 30, 9:22 pm, William Elliot <ma...@panix.com> wrote:
> On Sun, 30 Dec 2012, David C. Ullrich wrote:
> > William Elliot <ma...@panix.com> wrote:
>
> > >How long does an uncountable list with no duplicates,
> > >of infinite binary sequences (IBS) have to be to force
> > >the list to contain all the IBS's?

>
> > It's obvious that there is no length long enough to force this. A "list"
> > of length c (the cardinality of the set of infinite binary sequences)
> > need not contain all the sequences, and a list of length greater than c
> > must contain duplicates (and still need not contain all the sequences).

>
> Whoops.
>
> How long does an uncountable list have
> to be before it must contain a duplicate?


If you're still talking about lists of infinite binary sequences, then
the answer (assuming the axiom of choice) is the initial ordinal
omega_{alpha + 1} where alpha is the ordinal such that c =
aleph_{alpha}. But that's the answer given in the post you were
replying to, so I guess you're looking for some other kind of answer.



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