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Fatal Heyting Algebras and Forcing Persistent Sentences
 STUDIA LOGICA
"... Hamkins and Löwe proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra HZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting al ..."
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Hamkins and Löwe proved that the modal logic of forcing is S4.2. In this paper, we consider its modal companion, the intermediate logic KC and relate it to the fatal Heyting algebra HZFC of forcing persistent sentences. This Heyting algebra is equationally generic for the class of fatal Heyting algebras. Motivated by these results, we further analyse the class of fatal Heyting algebras.
STRUCTURAL CONNECTIONS BETWEEN A FORCING CLASS AND ITS MODAL LOGIC
"... Every definable forcing class Γ gives rise to a corresponding forcing modality, for which □Γ ϕ means that ϕ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [9] shows that if ZFC is consistent, t ..."
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Every definable forcing class Γ gives rise to a corresponding forcing modality, for which □Γ ϕ means that ϕ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [9] shows that if ZFC is consistent, then the ZFCprovably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2. In this article, we prove similarly that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CHpreserving forcing and others are each exactly S4.2; and the provably valid principles of ω1preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise.
On a question of Hamkins and Löwe (PRELIMINARY)
, 2008
"... Hamkins and Löwe asked whether there can be a model N of set theory with the property that N ≡ N[H] whenever H is a generic collapse of a cardinal of N onto ω. We obtain a lower bound, a cardinal κ with a κ +repeat point, for the consistency of such a model. We do not know how to construct such a m ..."
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Hamkins and Löwe asked whether there can be a model N of set theory with the property that N ≡ N[H] whenever H is a generic collapse of a cardinal of N onto ω. We obtain a lower bound, a cardinal κ with a κ +repeat point, for the consistency of such a model. We do not know how to construct such a model, under any assumption. We do construct, from a cardinal κ with o(κ) = κ +, a model N which satisfies the desired condition when H is the collapse of any successor cardinal. We also give a much weaker lower bound for this property. Joel Hamkins and Benedikt Löwe have asked, in connection with results reported in [1], whether there can be a model N of ZFC set theory such that N[H] ≡ N whenever H is the generic collapse of any cardinal onto ω. This note gives some partial results related to this question. In the positive direction we have the following partial result: Theorem 1. Suppose there is a cardinal κ with o(κ) = κ +.
Moving up and down in the generic multiverse
"... We investigate the modal logic of the generic multiverse which is a bimodal logic with operators corresponding to the relations “is a forcing extension of” and “is a ground model of”. The fragment of the first relation is the modal logic of forcing and was studied by the authors in earlier work. T ..."
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We investigate the modal logic of the generic multiverse which is a bimodal logic with operators corresponding to the relations “is a forcing extension of” and “is a ground model of”. The fragment of the first relation is the modal logic of forcing and was studied by the authors in earlier work. The fragment of the second relation is the modal logic of grounds and will be studied here for the first time. In addition, we discuss which combinations of modal logics are possible for the two fragments.