[FOM] Reinhardt cardinals
meskew at math.uci.edu
meskew at math.uci.edu
Thu Dec 15 13:13:44 EST 2011
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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