From: FGD <presbeis@my-deja.com>
Subject: Re: Continuum Hypotheisis
Date: Mon, 17 Jan 2000 17:53:52 GMT
Newsgroups: sci.math
Summary: [missing]

Just a few comments after some research and a few comments recieved by
email.

In article <85qrjl$b22$1@nnrp1.deja.com>,
  FGD <presbeis@my-deja.com> wrote:
> Aside: Does really PFA => 2^omega = omega_2? Though I've looked, I
> couldn't find a proof, if any. The usual forcing model of PFA does
> satisfy 2^omega = omega_2, but any suitably long countable support
> iteration of forcing will force 2^omega <= omega_2.

Yes. As someone pointed out to me by email, PFA => 2^omega = omega_2 is
a theorem due to Todorcevic and Velickovic. (I knew I had seen it
somewhere.)

> (Reflection too is a rather trivial fact,
> but I find it hard to think that such an important result is trivial.)

I've changed my mind, or almost: reflection itself is trivial as a model
theoretic statement (just think of the "algebraic" version), but it is
not trivial that we can prove it within ZF. That is, it's trivial from
outside, but not from inside.

> But you ask:
>
> >     Aside: does anyone know if any two such models agree
> >     on *all* Sigma^1_3 facts?  That would be more evidence
> >     for 0#.
>
> Which amounts to saying just that (for models of enough of ZFC+"0#
> exists"), doesn't it?

I think the answer is no. Someone pointed out to me that Jensen and
Jonsbraten's forcing argument by which they obtain a nonconstructible
Delta^1_3 set of natural numbers should go through over L[0#] instead of
L. (A simplified, but slightly sketchy, Jensen and Jonsbraten's appears
in Devlin and Jonsbraten's book "The Souslin Problem".)

> > Note that 0# exists if and only if
> >     \forall n (phi(n) <==> tau(n))
>
> (Where does the tau come from?)

Found it!

-- Frank


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