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Re: Uncountable Diagonal Problem
Posted:
Dec 30, 2012 4:32 AM
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On Dec 30, 2:47 am, William Elliot <ma...@panix.com> wrote:
> A list of length eta, is a function from the ordinals < beta
I would call that a list of length beta, not eta.
> to a set of items. Mega-sequence will be used as a synonym for list.
The usual term for such an object is "transfinite sequence".
> How long does a list without duplicates of infinite binary series (IBS)
> have to be to force the list to have every IBS?
I assume that by "series" you mean "sequence", and that by "infinite
binary series" you mean "binary sequence of length omega", in other
words, "element of {0, 1}^{omega}". And by "list without duplicates" I
suppose you mean "injective map from some ordinal". In plain language,
then, you seem to be asking the following:
"Let S = {0, 1}^{omega}. What is the least ordinal beta such that
every injective map from beta to S is surjective?"
More generally, let S be *any* infinite set. Trivially, if there is an
injective map from an ordinal beta to S, then there is an injective
map from that same ordinal beta to a proper subset of S. Even more
trivially, if there is no injective map from beta to S, then every
injective map from beta to S is surjective. Consequently, the least
ordinal beta from which every injective map from beta to S is
surjective is the same as the least ordinal beta for which there is no
injective map from beta to S; moreover, assuming the axiom of choice,
it's the same as the least ordinal beta such that |beta| > |S|.
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