Abstract
In this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK∗∗. We define this forcing by using a symmetry between MK∗∗ models and models of ZFC− plus there exists a strongly inaccessible cardinal (called SetMK∗∗). We develop a coding between β-models \(\mathcal {M}\) of MK∗∗ and transitive models M + of SetMK∗∗ which will allow us to go from \(\mathcal {M}\) to M + and vice versa. So instead of forcing with a hyperclass in MK∗∗ we can force over the corresponding SetMK∗∗ model with a class of conditions. For class-forcing to work in the context of ZFC− we show that the SetMK∗∗ model M + can be forced to look like \(L_{\kappa ^*}[X]\), where κ ∗ is the height of M +, κ strongly inaccessible in M + and X ⊆ κ. Over such a model we can apply definable class forcing and we arrive at an extension of M + from which we can go back to the corresponding β-model of MK∗∗, which will in turn be an extension of the original \(\mathcal {M}\). Our main result combines hyperclass forcing with coding methods of Beller et al. (Coding the universe. Lecture note series. Cambridge University Press, Cambridge, 1982) and Friedman (Fine structure and class forcing. de Gruyter series in logic and its applications, vol 3, Walter de Gruyter, New York, 2000) to show that every β-model of MK∗∗ can be extended to a minimal such model of MK∗∗ with the same ordinals. A simpler version of the proof also provides a new and analogous minimality result for models of second-order arithmetic.
Originally published in C. Antos, S.D. Friedman, Hyperclass Forcing in Morse-Kelley Class Theory. J. Symb. Logic 82(2), 549–575 (2017).
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Notes
- 1.
A detailed analyses on how even the Definability Lemma for class forcings can fail can be found in [7].
- 2.
Note that in ZFC minus Power Set the Bounding Principle does not follow from Replacement. This is used in [8], where he showed that in ZF− the different formulations of the Axiom of Choice are not equivalent. As for MK, work done in [5] shows that for example ultrapower constructions don’t work without first adding a version of Class-Bounding. For more information see [6].
- 3.
We can see here that it is vital to restrict ourselves to β-models in order to talk about minimal models of MK by comparing this to the situation in ZFC: There it also only makes sense to talk about minimal models containing a real for well-founded models (and not for ill-founded models). So by making the transformation from MK to SetMK we have to restrict ourselves to β-models.
- 4.
- 5.
See [4], Theorem 7.5, p. 142.
- 6.
See [4, Chapter 4].
- 7.
This follows from an property called diagonal distributivity (see [4], p. 37).
References
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Acknowledgements
The first author wants to thank the Austrian Academy of Sciences for their generous support through their Doctoral Fellowship Program. Both authors are grateful to the John Templeton Foundation for its generous support through Grant ID 35216, which supported the preparation of this article.
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Antos, C., Friedman, SD. (2018). Hyperclass Forcing in Morse-Kelley Class Theory. In: Antos, C., Friedman, SD., Honzik, R., Ternullo, C. (eds) The Hyperuniverse Project and Maximality. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-62935-3_2
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