## THE MODAL LOGIC OF FORCING (2007)

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Citations: | 4 - 2 self |

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@MISC{David07themodal,

author = {Joel David and Hamkins and Benedikt L Öwe},

title = {THE MODAL LOGIC OF FORCING},

year = {2007}

}

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### Abstract

Abstract. A set theoretical assertion ψ is forceable or possible, written ♦ ψ, if ψ holds in some forcing extension, and necessary, written � ψ, ifψ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2. 1.

### Citations

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123 |
Maarten de Rijke, and Yde Venema, Modal logic, Cambridge Tracts
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(Show Context)
Citation Context ...he axioms under modus ponens, substitution and necessitation. This list is not exhaustive, as there are continuum many modal theories above S4.2 that are not listed. We refer the reader to [CZ97] and =-=[BdRV01]-=- for excellent developments of modal logic, including the analysis of these and many other theories.s6 JOEL DAVID HAMKINS AND BENEDIKT L ÖWE Some Common Modal Theories S5 = S4 + 5 S4W5 = S4 + W5 S4�3 ... |

112 | The Logic of Provability - Boolos - 1993 |

54 |
Provability interpretations of modal logic
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(Show Context)
Citation Context ... The authors would like to thank Nick Bezhanishvili (Amsterdam), Dick de Jongh (Amsterdam), Marcus Kracht (Los Angeles CA), and Clemens Kupke (Amsterdam) for sharing their knowledge of modal logic. 1 =-=[Sol76]-=-; for a survey of the result and the subsequent development of the field of provability logic, see also [JdJ98]. 1s2 JOEL DAVID HAMKINS AND BENEDIKT L ÖWE set theoretical truths in a way that can ofte... |

39 |
Modal Logic, volume 35 of Oxford Logic Guides
- Chagrov, Zakharyaschev
- 1997
(Show Context)
Citation Context ...y closing the axioms under modus ponens, substitution and necessitation. This list is not exhaustive, as there are continuum many modal theories above S4.2 that are not listed. We refer the reader to =-=[CZ97]-=- and [BdRV01] for excellent developments of modal logic, including the analysis of these and many other theories.s6 JOEL DAVID HAMKINS AND BENEDIKT L ÖWE Some Common Modal Theories S5 = S4 + 5 S4W5 = ... |

27 |
Applications of Kripke models
- Smorynski
(Show Context)
Citation Context ...rn of buttons and switches, any larger pattern of buttons and any pattern of switches is possible. The main technique in our proofs of Lemmas 6.2 and 6.3 appears to be very reminiscent of Smoryński’s =-=[Smo70]-=- proof of de Jongh’s theorem [dJ70] on Heyting’s Arithmetic. Lemma 7.3. If F is a finite pre-lattice, w0 ∈ F and Λ ⊇ S4 is a modal theory consistent with a sufficiently large independent family of but... |

9 |
The maximality of the intuitionistic predicate calculus with respect to Heyting's Arithmetic
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(Show Context)
Citation Context ...er pattern of buttons and any pattern of switches is possible. The main technique in our proofs of Lemmas 6.2 and 6.3 appears to be very reminiscent of Smoryński’s [Smo93] proof of de Jongh’s theorem =-=[dJ70]-=- on Heyting’s Arithmetic. Lemma 7.3. If F is a finite pre-lattice, w0 ∈ F and Λ ⊇ S4 is a modal theory consistent with a sufficiently large independent family of buttons and switches, then Λ is consis... |

6 |
Infinitary combinatorics and modal logic
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(Show Context)
Citation Context ...rinciple, a new forcing axiom, with related work in [Lei04] and [HW05]. An alternative but related connection between modal logic and forcing was explored by Fitting and Smullyan in [SF96], and Blass =-=[Bla90]-=- provides an interpretation of modal logic in set theory that is not directly related to forcing. These modal operators, of course, are eliminable in the language of set theory, because their meaning ... |

5 | The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal - Hamkins, Woodin |

4 | Set Theory. Spring Monographs in Mathematics - Jech - 2003 |

3 |
Hamkins. A simple maximality principle
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(Show Context)
Citation Context ...lds in all forcing extensions. The modal notation ✸ ϕ and ✷ϕ expresses, respectively, that ϕ is possible or necessary. This forcing interpretation of modal logic was introduced by the first author in =-=[Ham03]-=- in connection with the Maximality Principle, a new forcing axiom, with related work in [Lei04] and [HW05]. An alternative but related connection between modal logic and forcing was explored by Fittin... |

3 |
Consistency strengths of maximality principles
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(Show Context)
Citation Context ... possible or necessary. This forcing interpretation of modal logic was introduced by the first author in [Ham03] in connection with the Maximality Principle, a new forcing axiom, with related work in =-=[Lei04]-=- and [HW05]. An alternative but related connection between modal logic and forcing was explored by Fitting and Smullyan in [SF96], and Blass [Bla90] provides an interpretation of modal logic in set th... |

3 |
Japaridze and Dick De Jongh, The logic of provability, Handbook of proof theory
- Giorgi
- 1998
(Show Context)
Citation Context ...ngeles, CA), and Clemens Kupke (Amsterdam) for sharing their knowledge of modal logic. 1 [Sol76]; for a survey of the result and the subsequent development of the field of provability logic, see also =-=[JdJ98]-=-. 1793 c○2007 American Mathematical Society Reverts to public domain 28 years from publications1794 JOEL DAVID HAMKINS AND BENEDIKT LÖWE all models of set theory, related by forcing, as an enormous Kr... |

1 |
Japaridze and Dick de Jongh. Handbook of Proof Theory, chapter VII: The Logic of Provability
- Giorgi
- 1998
(Show Context)
Citation Context ...Angeles CA), and Clemens Kupke (Amsterdam) for sharing their knowledge of modal logic. 1 [Sol76]; for a survey of the result and the subsequent development of the field of provability logic, see also =-=[JdJ98]-=-. 1s2 JOEL DAVID HAMKINS AND BENEDIKT L ÖWE set theoretical truths in a way that can often be carefully controlled. The method has become a fundamental tool in set theory. Because the ground model V h... |

1 |
Set Theory and the Continuum Problem, volume 34 of Oxford Logic Guides
- Smullyan, Fitting
- 1996
(Show Context)
Citation Context ...h the Maximality Principle, a new forcing axiom, with related work in [Lei04] and [HW05]. An alternative but related connection between modal logic and forcing was explored by Fitting and Smullyan in =-=[SF96]-=-, and Blass [Bla90] provides an interpretation of modal logic in set theory that is not directly related to forcing. These modal operators, of course, are eliminable in the language of set theory, bec... |

1 |
Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, chapter V. Applications of Kripke Models
- Smoryński
- 1993
(Show Context)
Citation Context ...rn of buttons and switches, any larger pattern of buttons and any pattern of switches is possible. The main technique in our proofs of Lemmas 6.2 and 6.3 appears to be very reminiscent of Smoryński’s =-=[Smo93]-=- proof of de Jongh’s theorem [dJ70] on Heyting’s Arithmetic. Lemma 7.3. If F is a finite pre-lattice, w0 ∈ F and Λ ⊇ S4 is a modal theory consistent with a sufficiently large independent family of but... |

1 |
Modal Logic, volume35ofOxford Logic Guides, Oxford Science Publications
- Chagrov, Zakharyaschev
- 1997
(Show Context)
Citation Context ...y closing the axioms under modus ponens, substitution and necessitation. This list is not exhaustive, as there are continuum many modal theories above S4.2 that are not listed. We refer the reader to =-=[CZ97]-=- and [BdRV01] for excellent developments of modal logic, including the analysis of these and many other theories.sS5 = S4 +5 S4W5 = S4 +W5 S4.3 = S4 + .3 S4.2.1 = S4 + .2+M S4.2 = S4 + .2 S4.1 = S4 +M... |

1 | The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal - HughWoodin |

1 |
On ∇-model of set theory
- Vopěnka
- 1965
(Show Context)
Citation Context ...s the forcing extension as an actual structure (although the new ground model may not be isomorphic to M and may not even be well founded). Vopěnka seems to have been the first to do forcing this way =-=[Vop65]-=-.sTHE MODAL LOGIC OF FORCING 1795 Main Definition 1. A modal assertion ϕ(q0,...,qn)isavalid principle of forcing if for all sentences ψi in the language of set theory, ϕ(ψ0,...,ψn) holds under the for... |