[FOM] Reinhardt cardinals

[Andrew Brooke-Taylor]

Perhaps it is worth highlighting a subtlety in Monroe's Open Question (2).
 Kunen's inconsistency proof is for a universe V that is a model of second
order ZFC, at least as far as the class j is concerned: crucial use is made
of the Axiom Schema of Replacement for formulas involving the predicate j.
 If the "theory with existing proper classes" does not satisfy much
Replacement with respect to the proper classes, then solutions are known.
 For example, if 0# exists, then L with the 0# embedding, appropriately
formalised, can be seen as an example for Monroe's question (2) - of
course, L is very far from satisfying Replacement for formulas with a
predicate for 0#.  Note however that NBG does require Replacement for class
functions, so such an example could not be so easily constructed in that
setting.

Paul Corazza has extensively studied this situation, proposing the
"Wholeness Axiom", in which there is an elementary embedding j from V to V
such that
V satisfies the Axiom Schema of Separation for formulas with a predicate
for j, but not Replacement; this is consistency-wise below the existence of
an I3 embedding, but implies the existence of a cardinal that is n-huge for
all finite n.  His papers are available on his website (
http://pcorazza.lisco.com/mathPublications.html) and a good first port of
call might be "The spectrum of elementary embeddings j:V-->V", Annals of
Pure and Applied Logic 139 (2006) pp 327-399 (
http://www.sciencedirect.com/science/article/pii/S0168007205000953).

Best wishes,
Andrew Brooke-Taylor

On 17 December 2011 03:54, <meskew at math.uci.edu> wrote:

> > Can you give a reference for Suzuki's theorem?
> >
> > Thanks.
> >
> > -- Bob Solovay
>
>
> Here is the citation:
>
> "No Elementary Embedding from V into V is Definable from Parameters,"
> Akira Suzuki, The Journal of Symbolic Logic , Vol. 64, No. 4 (Dec., 1999),
> pp. 1591-1594.
>
> It is also discussed in this recent paper by Hamkins et al:
> http://arxiv.org/abs/1106.1951
>
> The idea is simple.  For \phi(x,y,z), let \kappa be least such that for
> some z, \phi(x,y,z) defines a Sigma_1 elementary embedding into V with
> critical point \kappa.  Pick some witness and apply its embedding, noting
> that the above statement is preserved.
>
> Best,
> Monroe
>
>
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