Re: the I3 axiom

Rupert wrote:

> "Robert E. Beaudoin" <[EMAIL PROTECTED]> wrote in message news:<[EMAIL PROTECTED]>...
>
> > Rupert wrote:
> >
> > > I have been trying to understand why Kunen's proof doesn't show that
> > > there can't be a nontrivial elementary embedding from a rank to
> > > itself.
> > >
> > > If I have it right, the point is that if j is a nontrivial elementary
> > > embedding V_lambda->V_lambda, kappa the least ordinal moved, then the
> > > limit of {kappa,j(kappa),j(j(kappa)),...} needn't be less than lambda,
> > > it might equal lambda.
> > >
> > > So Kunen's result shows that such lambda if they exist have cofinality
> > > omega.
> > >
> > > Is this right?
> >
> > Almost but not quite.  Let zeta be the supremum of
> > {kappa, j(kappa), j(j(kappa)), ...}.  Then zeta has countable
> > cofinality (of course) and is less than or equal to lambda.
> > Kunen's argument that there is no elementary embedding from V to V
> > (assuming AC) in fact shows more, namely that lambda is at most
> > zeta + 1.  But I don't think it is known that lambda = zeta + 1 can
> > be ruled out; i.e. existence of an elementary embedding from a
> > successor (of an ordinal of countable cofinality, of course) rank
> > to itself has not to my knowledge been shown inconsistent with ZFC.
> >
> > Robert E. Beaudoin
>
>
> Apparently V_zeta is a model of ZFC. How do you prove that?

Since kappa is the critical point of j, kappa is strongly
inaccessible.  So zeta is a strong limit cardinal.  That makes it
pretty clear that M = V_{zeta} is a model of all of ZFC except
perhaps the collection scheme.  So suppose F: M -> M is definable
over M with parameter p, and a is in M; one must show F"a is
contained in some element of M.  For each cardinal eta in
{kappa, j(kappa), j(j(kappa)), ...} there is an elementary
embedding i: M -> M with critical point eta (obtained by finitely
many self-applications of j) and such that the supremum of
{eta, i(eta), i(i(eta)), ...} is zeta.  For sufficiently large
eta both p and a have rank less than eta, whence i fixes p, a,
and each element of a.  For any x in a the rank of F(x) is thus
fixed by i, and so is less than eta.  So F"a is a subset of
V_{eta} and we are done.

Hope that helps.

Robert E. Beaudoin