2015/10/14 by Carolin Antos, Antos, Carolin, Sy‐David Friedman +2
Computer Science · Mathematics · #03E40 #03E70 #03Exx #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO #msc:03E40 #msc:03E70 #msc:03Exx
paper · pdf · doi:10.48550/arxiv.1510.04082
arxiv created 2015/10/14 · openalex publication_date 2015/10/14 · arxiv updated 2015/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 M of MK** and transitive models M+ of SetMK** which will allow us to go from 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 Lκ^*[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 M. Our main result combines hyperclass forcing with coding methods of [BJW82] and [Fri00] 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.