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 < 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|.