[FOM] The Kunen inconsistency and definable classes

[Sam Roberts]

Dear all,

There is a tension between (1) interpreting proper class talk in set theory
as talk about first-order formulas and satisfaction; and (2) taking it to
be an interesting and non-trivial result that there is no (non-trivial)
elementary embedding from V into V and/or taking it to be an open question
whether there can be such an e.e. in the absence of choice. Basically,
there is a very simple proof that there can be no definable e.e. from V
into V (see Suzuki (1999)).

This tension was recently highlighted by Hamkins, Kirmayer, and Perlmutter
(2012). There, the resolution was to give up on (1), since accepting it
``does not convey the full power of the [Kunen's] theorem" (p. 1873). But
this is perhaps the only place I've seen this issue addressed. For
instance, Kanamori seems to hold both (1) and (2) in  The Higher Infinite:
``By “class” in the ZFC context is meant definable class,... [x \in M]* *is
merely [a] facon de parler" (p. 33); and ``[t]he following unresolved
question [i.e. whether there could be an e.e. from V into V in the absence
of choice] is therefore of foundational interest" (p. 324).

My question is: what do other set theorists think of this tension, and how
do they prefer to resolve it?

All the best,
Sam Roberts

Hamkins, J., Kirmayer, G., Perlmutter, N. (2012) ``Generalizations of the
Kunen inconsistency". Annals of Pure and Applied Logic, 163, 1872–1890.

Suzuki, A. (1999) No elementary embedding from V into V is definable from
parameters. Journal of Symbolic Logic 64, 1591-1594.
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