Re: Aleph One Sets

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 10/20/04 

Date: Wed, 20 Oct 2004 11:59:55 -0500

Apollo Hogan wrote:
> In article <cl39iv0r3s@drn.newsguy.com>,
> Daryl McCullough <daryl@atc-nycorp.com> wrote:

>>Here's a cardinality question for you (the answer *might* be aleph_1)
>>
>>Define an equivalence class on sets of reals as follows: R1 ~~ R2 if
>>there exists an order-preserving bijection between R1 and R2. What is
>>the cardinality of the set of equivalence classes of ~~?
>
>
> Well, if 2^aleph_0 = aleph_beta, then there will be at least beta different
> equivalence classes (one of each cardinality), so, for example, in a model
> where 2^aleph_0 = aleph_(omega_2), there are (at least) aleph_2 different
> equivalence classes.

Nice!

But one can do better--it's not hard to show directly that there are
at least 2^{aleph_0} distinct classes. Embed length-omega strings of
zeroes and ones thusly: For each 0, put a copy of omega in
an interval; for each 1, a copy of omega* (that's omega reversed).

Here's what I'm wondering: Can you actually embed 2^{2^aleph_0}?
I don't immediately see why or why not.

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