[FOM] Large chains in the powerset of the reals

George Barmpalias georgeb at maths.leeds.ac.uk
Tue Oct 14 00:56:29 EDT 2008

Wis Comfort has pointed out the following answer to my question:

the existence of chains in the power set of reals under inclusion, of size
the cardinality of the power set of reals is independent of ZFC.

This is essentially proved (among other things) in

Baumgartner, James E. Almost-disjoint sets, the dense
set problem and the partition calculus. Ann. Math. Logic 9 (1976), no. 4,

and is explicitly stated as corollary 1.3 in

W. W. Comfort and D. Remus
Long chains of Hausdorff topological group topologies,
Journal of Pure and Applied Algebra 70 (1991) 53-72

In these papers, many related problems are studied.
For further information I include my correspondence with Prof. Comfort.

Dear Professor Barmpalias,
     I didn't know about the FOM mailing list but I had e-mail
yesterday from Carol Wood. I append our exchange below.
I don't know if it's conventional to post responses on that list
---I haven't done that, and if I were writing "for the public" I
might take little more care and more time than in this quick
message to Carol---but for what it's worth I pass it on to you.
     I would be pleased to correspond further with you about
                                                     Wistar Comfort
cc: Carol Wood

>Date: Wed, 08 Oct 2008 14:54:27 -0400
>To: Carol Wood <cwood at wesleyan.edu>
>From: Wistar Comfort <wcomfort at wesleyan.edu>
>Subject: Re: from the FOM mailing list
>>  Is the independence from ZFC of the following statement known?
>>  "There are chains in the power set of reals under inclusion, which have
>>  the same cardinality as the power set of reals."
>>  Can you provide any references?
>  Hi, Carol, I'm pretty sure the one-word answer is Yes and Yes.
>The best reference is Baumgartner, Almost disjoint sets, the dense set
>problem and the partition calculus,
>Annals of Math Logic 10 (1976), 401-439. Here are some more details:
>      Let C(\kappa,\lambda) (my notation, I think but I'm not
sure---anyway I
>think of "C" as meaning "chain") be the statement that the power set
>\sP(\kappa) (under
>inclusion) contains a chain of length \lambda. Then Baumgartner shows
>      (a) C(2^\omega,2^{(2^\omega)}) fails in some models of ZFC;
>      (b) C(2^\kappa,2^{\kappa^+)}) holds [ZFC] for all \kappa\geq\omega;
>      (c) C(\kappa,\kappa^+) holds [ZFC] for all \kappa\geq\omega;
>      (d) Under GCH, C(\kappa,2^\kappa) holds for all \kappa\geq\omega.
>Statement (a) is the question you posed I believe. Of course (d) follows
>from (a).
>      It's amusing that if \kappa+=2^\kappa then C(\kappa,2^\kappa) holds
>(that's from (b) above), but also C(\kappa^+,2^{\kappa^+}) holds. In other
>from one isolated "instance of GCH" one gets a chain of maximal length for
>two different cardinals.
>      The real theorem in this direction, from which all  known positive
> results flow
>I think, is that C(2^{<\lambda}, 2^\lambda) holds for all \lambda\geq\omega.
>      Of course Baumgartner does much more than what I cited above, for
> example
>he considers the more subtle condition C(|kappa,\lambda,\mu) meaning that
>in the power set \sP(\kappa) there is a chain of length \lambda of which
>member has cardinality \mu.
>      My topological contributions in this direction are with Dieter
Remus in
>J. Pure and Applied Algebra 70 (1991), 53--72 and Topology and Its
>75 (1997), 51-79. See also Berarducci, Dikranjan, Forti and Watson, J.
>Pure and
>Applied Algebra 126 (1998), 19--49.
>                                                                      Best
> wishes,
>                                                                        Wis

W. Wistar Comfort                          phone: 860-685-2632 (office)
Dept. of Mathematics                    FAX: 860-685-2571 (office)
Wesleyan University                        phone: 860-434-7971 (home)
Middletown, CT 06459

Dear Professor Barmpalias,
    Thank you for the follow-up message. I am not a professional set-theorist
but I want to hold to my earlier claim(s) until proven wrong. I would be
if you decide to post my earlier e-mail to you on that web-site, maybe along
with this e-mail. Let me make some additional remarks.
    Let us say (my notation, maybe not optimal) for infinite cardinals
$\kappa$ and
$\lambda$ that C(\kappa,\lambda)$ means: there is in the power set
ordered by inclusion, a chain of length $\lambda$. Then, $C(\omega,2^\omega)$
is well known and easy to prove. [With each real number $r$, associate the
set of rational numbers $\{q\in Q:q<r\}$. That gives a chain of length $|R|
in the power set $\sP(Q)$.]
    As Prof Marker indicated in the e-mail you sent me, the same proof works
to show
that in any linearly ordered set $L$ with a dense subset $D$, one would have
the condition $C(|D|,|L|)$. In fact the paper I cited by Baumgartner in
of Math
Logic 10 (1976), 401-439 shows (there's a lot of neat and difficult
mathematics in that
paper, but this theorem is pretty easy) that the condition
equivalent to the existence of a linearly ordered set $L$ such that $|L|
=\lambda$ and
$L$ has a dense set $D$ such that $|D|=\kappa$. [See his Theorem 2.1
there. In
careful survey of this issue it would be appropriate, as Baumgartner makes
in his discussion, to mention the names and work of Sierpinski, Tarski, J.
and W. Mitchell.]
    I mentioned earlier that $C(2^{<\kappa},2^\kappa)$ always holds (not hard
show), so always $C(\kappa,\kappa^+)$ holds in ZFC for all (infinite)
It is then immediate that if $\kappa^+=2^\kappa$, then $C(\kappa,2^\kappa)$
and also $C(\kappa^+,2^{\kappa^+})$ hold. That is all clear in Baumgartner's
paper. For a "direct straightforward" proof of that, and some other simple
facts which demand no new ideas, see my paper with Remus J. Pure and
Applied Algebra 70 (1991), 53--72, especially Section 1.
    The model-theoretic work in Baumgartner's paper goes to show that
$C(\kappa,2^\kappa)$ can fail for many $\kappa$. For example, he states
specifically that $C(\aleph_1,2^{\aleph_1})$ can fail.
    You were asking about the failure of $C(\kappa,2^\kappa)$ when
$\kappa=c=|R|$. That consistent failure is noted explicitly in my paper with
Remus (Corollary 1.3), again using this theorem of Baumgartner: If
$\kappa=2^\alpha$ is singular, and if there is a $\cf(\kappa)$-sequence
$\kappa_\xi$ such that $\kappa=\sup\{\kappa_\xi:\xi<\cf(\kappa)\} such
that $\kappa_\xi<\kappa_\eta<\kappa<\2^{\kappa_\xi}<2^{\kappa_\eta}\}$,
then $C(\kappa,2^\kappa)$ fails. [Remus and I again give a short proof
of that Theorem of Baumgartner, see our Theorem 1.2.]  Then,
$C(c,2^c)$ fails in a model with $c=\aleph_{\aleph_1}$ in which also
$2^{\aleph_\xi}<2^{\aleph_\eta}$ whenever $\omega<\xi<\eta<\aleph_1$.
   In writing that paper with Remus, I was interested in whether
$C(c,2^c)$ could fail in some model where $c$ is regular. There, I
enlisted some help from some genuine set theorists, Alan Dow and
Stevo Todorcevic. They both pointed out that the answer is Yes,
that condition can fail even with $c=\aleph_2$, further that follows
from a careful reading of Baumgartner's paper (see my paper with
Remus for details about that consistent failure).
    I want to mention, to give credit where it is due, that in these set-
theoretic portions of my paper with Remus, we profited from some
good conversations with Istvan Juhasz (see our Remark 1.4).
    My paper with Remus, also its sequel which I mentioned in my earlier
e-mail and the subsequent improvements and clarifications by
Berarducci, Dikranjan, Forti and Watson, are concerned with the
(possible) existence of long chains of totally bounded group topologies
on abelian groups. An earlier theorem of mine with Kenneth A. Ross
(Fundamenta Math. 55 (1964), 283--291) essentially reduces the
question of the existence of such chains of pre-assigned length to the
existence of chains of such length in an appropriate power set---a
strictly set-theoretic question.
    These remarks are quite informal, typed out in a single short sitting.
I feel some confidence, but that confidence is weakened a bit by some
of the statements you have sent me from prominent mathematicians.
If you find my analysis faulty, I would appreciate hearing from you (or
any one else) about that.
   I don't know about FOM, I leave it to you (and I thank you for it) to
post these e-mails over there if you think that could be useful.
                                                       Wistar Comfort

W. Wistar Comfort                           phone: 860-685-2632 (office)
Dept. of Mathematics                     FAX: 860-685-2571 (office)
Wesleyan University                        phone: 860-434-7971 (home)
Middletown, CT 06459

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