[FOM] The Kunen inconsistency and definable classes

Monroe Eskew meskew at math.uci.edu
Fri May 3 15:20:31 EDT 2013

I raised this point some time ago on the FOM list.  My personal opinion is that we should officially keep (1) and interpret Kunen's theorem saying that in ZFC, there cannot exist an elementary j : V_{\kappa+2} \to V_{\kappa+2}.


On May 2, 2013, at 11:46 PM, Sam Roberts <srober21 at mail.bbk.ac.uk> wrote:

> 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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