```Date: Dec 30, 2012 4:32 AM
Author: Butch Malahide
Subject: Re: Uncountable Diagonal Problem

On Dec 30, 2:47 am, William Elliot <ma...@panix.com> wrote:> A list of length eta, is a function from the ordinals < betaI 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 "infinitebinary series" you mean "binary sequence of length omega", in otherwords, "element of {0, 1}^{omega}". And by "list without duplicates" Isuppose 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 thatevery injective map from beta to S is surjective?"More generally, let S be *any* infinite set. Trivially, if there is aninjective map from an ordinal beta to S, then there is an injectivemap from that same ordinal beta to a proper subset of S. Even moretrivially, if there is no injective map from beta to S, then everyinjective map from beta to S is surjective. Consequently, the leastordinal beta from which every injective map from beta to S issurjective is the same as the least ordinal beta for which there is noinjective 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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