[FOM] Reinhardt cardinals
Robert Solovay
solovay at gmail.com
Fri Dec 16 03:05:46 EST 2011
Can you give a reference for Suzuki's theorem?
Thanks.
-- Bob Solovay
On Dec 15, 2011 8:59 PM, <meskew at math.uci.edu> wrote:
> I would like to clear up a misconception about Reinhardt cardinals. I had
> this misconception until recently.
>
> In the current edition his book, _The Higher Infinite_, Kanamori states
> the following "unresolved question" on page 324:
>
> Is it provable in ZF that there is no nontrivial elementary embedding j :
> V \to V?
>
> I claim that this question, when phrased in this way, has in fact been
> resolved.
>
> According to ZF, there are no proper classes. Classes are merely informal
> shorthand for talk about particular formulas with parameters. If the
> above question is to be about ZF, it must really be the following:
>
> Is the following schema provable in ZF: For all A, \phi(x,y,A) does not
> define an elementary embedding from V to V?
>
> Due to work of Gaifman, the concept "elementary embedding from V to V" can
> be expressed in the language of ZF since "Sigma_1 elementary
> embedding" in fact suffices.
>
> If we do not ask this question in schema form, then it is not even in the
> language of ZF, so the answer to whether the statement is provable in ZF
> is trivially, "no."
>
> Now the schema form of this question was fully answered by Suzuki in 1999.
> He published a simple diagonal argument that no nontrivial j: V \to V
> can definable from parameters, assuming only that V satisfies ZF.
>
> Still open are the following questions:
>
> 1) Can there be a "weak Reinhardt cardinal" in ZF, the critical point of a
> set function j : V_{\lambda+n} \to V_{\lambda+n}, where n>1?
> 2) In a theory with existing proper classes such as NBG without local
> choice, can there exist a class J which is an elementary map for
> the language of ZF?
>
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