# The modal logic of forcing

@article{Hamkins2005TheML,
title={The modal logic of forcing},
author={J. Hamkins and B. Loewe},
journal={Journal of Immunology},
year={2005}
}
• Published 2005
• Computer Science, Biology, Mathematics, Geology
• Journal of Immunology
A set theoretical assertion psi is forceable or possible, written lozenge psi, if psi holds in some forcing extension, and necessary, written square psi, if psi 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.

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#### References

SHOWING 1-10 OF 42 REFERENCES
Provability interpretations of modal logic
We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and letExpand
Infinitary Combinatorics and Modal Logic
• A. Blass
• Mathematics, Computer Science
• J. Symb. Log.
• 1990
We show that the modal propositional logic G , originally introduced to describe the modality “it is provable that”, is also sound for various interpretations using filters on ordinal numbers, forExpand
Consistency Strengths of Modified Maximality Principles
The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then itExpand
Consistency Strengths of Modified Maximality Principles
The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then itExpand
A simple maximality principle
• J. Hamkins
• Mathematics, Computer Science
• Journal of Symbolic Logic
• 2003
Abstract In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extensionExpand
The logic of provability
1. GL and other systems of propositional modal logic 2. Peano arithmetic 3. The box as Bew(x) 4. Semantics for GL and other modal logics 5. Completeness and decidability of GL and K, K4, T, B, S4,Expand
Modal logic
This paper surveys the main concepts and systems of modal logic. It shows how the tree or tableau method provides a simple and easily comprehensible decision procedure for systems such as K, T, S4Expand
Metamathematical investigation of intuitionistic arithmetic and analysis
Intuitionistic formal systems.- Models and computability.- Realizability and functional interpretations.- Normalization theorems for systems of natural deduction.- Applications of Kripke models.-Expand
The logic of the provability
• Mathematics, Computer Science
• 1998
This chapter is dedicated to the memory of George Boolos. From the start of the subject until his death on 27 May 1996 he was the prime inspirer of the work in the logic of provability.
The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinal
• Mathematics, Computer Science
• Math. Log. Q.
• 2005
The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing,Expand