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I would like to have something like a linear order on classes, such that every instantiated predicate of classes has a minimal instance in that order. For my purposes, it is fine to assume V=L for sets.

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    $\begingroup$ I'm not quite clear as to what you mean. If you assume $V=L$ then you have a definable well-ordering of the sets, therefore every non-empty class must have a least element. $\endgroup$ – Asaf Karagila Jan 7 '15 at 19:32
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    $\begingroup$ @AsafKaragila I want well ordering not in classes, but of classes. $\endgroup$ – Colin McLarty Jan 7 '15 at 20:06
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In the context of models of second-order arithmetic, Mostowski showed that starting with the theory $Z_2$ (the second-order arithmetic analogue of Kelley-Morse) together with the scheme of dependent choices, one can use forcing to add a well-ordering relation on classes so that the full second-order comprehension scheme continues to hold in the language with the new relation. The argument appeared in his paper "Models of second order arithmetic with definable Skolem functions" [Fund. Math. 75 (1972), 223–234] and was later corrected in "Erratum to the paper `Models of second order arithmetic with definable Skolem functions'" [Fund. Math. 84 (1974), 173]. Initially, Mostowski used only the choice scheme to prove the result.

Mostowski's argument generalizes to the theory Kelley-Morse together with the scheme of dependent choices. The scheme of dependent choices states that if a second-order definable relation has no terminal nodes, then it has an $\omega$-chain. The forcing is supposed to be akin to the forcing argument that a global well-ordering class can be added to a model of ZFC, but in this case the condition are classes, and so we have a hyperclass forcing (in Sy Friedman's terminology). I only know about this construction from conversations with Ali Enayat, who will probably give many more details when he is around.

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  • $\begingroup$ Terrific. I will take some time to absorb this. Since I really want this in a weak fragment of ZFC, the use of $Z_2$ is very encouraging. $\endgroup$ – Colin McLarty Jan 7 '15 at 20:25
  • $\begingroup$ @ColinMcLarty I think the key to making the argument work is having full second-order comprehension (which GB doesn't have) plus the dependent choice scheme for classes. This is, I think, what is needed for hyperclass forcing to preserve comprehension in the extended language. But I am hoping Ali will be here to say more. $\endgroup$ – Victoria Gitman Jan 7 '15 at 22:08
  • $\begingroup$ A beautiful result. I fear it sinks my strategy for the problem that led to this question, but maybe that failure points the way to what will work. $\endgroup$ – Colin McLarty Jan 9 '15 at 20:26

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