From: "Daniel Giaimo" <dgiaimo@ix.netcom.com>
Subject: Re: Set Theory Question
Date: Mon, 3 Jan 2000 14:42:37 -0800
Newsgroups: sci.math
Summary: regular cardinals and cardinalities of small subsets

Fred Galvin <galvin@math.ukans.edu> wrote in message
news:Pine.LNX.4.10.9912311851000.22743-100000@titania.math.ukans.edu...
> On Fri, 31 Dec 1999, Daniel Giaimo wrote:
>
> >     Let S be any infinite set.  Let T be the set of all subsets P
> > of S such that card(P) < card(S).  Is it true, and if so how would
> > you prove, that card(T) = card(S)?
>
> Not necessarily, as I pointed out in my earlier reply. On the other
> hand, assuming GCH, it is true for most values of card(S); namely, it
> is true when card(S) is a regular cardinal. More generally, if S is
> any infinite set, and if T' is the set of all subsets P of S such that
> card(P) < cf(card(S)), then card(T') = card(S).

    I'm not sure I see why card(S) >= card(T').  (Clearly, card(S) <=
card(T')
as T' contains the singleton subsets of S.)

--Daniel Giaimo

==============================================================================

From: "Brian M. Scott" <BMScott@stratos.net>
Subject: Re: Set Theory Question
Date: Mon, 03 Jan 2000 21:17:23 -0500
Newsgroups: sci.math

Daniel Giaimo wrote:

> Fred Galvin <galvin@math.ukans.edu> wrote

[...]

> > Not necessarily, as I pointed out in my earlier reply. On the other
> > hand, assuming GCH, it is true for most values of card(S); namely, it
> > is true when card(S) is a regular cardinal. More generally, if S is
> > any infinite set, and if T' is the set of all subsets P of S such that
> > card(P) < cf(card(S)), then card(T') = card(S).

>     I'm not sure I see why card(S) >= card(T').  (Clearly, card(S) <=
> card(T') as T' contains the singleton subsets of S.)

Well-order S in type card(S), S = {s(i) : i < card(S)}.  Let 
k = cf(card(S)), and let (m(i) : i < k) be an increasing sequence 
of cardinals cofinal in card(S).  If P is in T', then P is a
subset of S(j) = {s(i) : i < m(j)} for some j < k.  By GCH we 
have card(powerset(S(j))) = m(j)+ <= m(j+1) < card(S), so
card(T') <= k card(S) = card(S).

Brian M. Scott