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.

  • 1
    $\begingroup$ 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
  • 4
    $\begingroup$ @AsafKaragila I want well ordering not in classes, but of classes. $\endgroup$ – Colin McLarty Jan 7 '15 at 20:06

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.