
Re: Uncountable Diagonal Problem
Posted:
Dec 30, 2012 4:32 AM


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. Megasequence 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.

