Related papers: The modal logic of $\sigma$-centered forcing and r…
What are the most general principles in set theory relating forceability and truth? As with Solovay's celebrated analysis of provability, both this question and its answer are naturally formulated with modal logic. We aim to do for…
The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, interpreting necessity as "in all forcing extensions" and possibility as "in some forcing extension". In this modal…
I introduce a new family of axioms extending ZFC set theory, the $\Sigma_n$-correct forcing axioms. These assert roughly that whenever a forcing name $\dot{a}$ can be forced by a poset in some forcing class $\Gamma$ to have some $\Sigma_n$…
We study principles of the form: if a name $\sigma$ is forced to have a certain property $\varphi$, then there is a ground model filter $g$ such that $\sigma^g$ satisfies $\varphi$. We prove a general correspondence connecting these name…
Solovay's arithmetical completeness theorem states that the modal logic of provability coincides with the modal logic $\mathbf{GL}$. Hamkins and L\"owe studied the modal logical aspects of set theoretic multiverse and proved that the modal…
We determine the ZF-provable modal logic of the modality $\Box_{\mathrm{sym}}$, where $\Box_{\mathrm{sym}}\varphi$ means '$\varphi$ holds in every finite symmetry-preserving iteration' of the symmetric method. We prove that the exact logic…
In this article we adapt the existing account of class-forcing over a ZFC model to a model $(M,\mathcal{C})$ of Morse-Kelley class theory. We give a rigorous definition of class-forcing in such a model and show that the Definability Lemma…
We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct a proper generic extension and show that the latter also satisfies…
The class forcing theorem, which asserts that every class forcing notion $\mathbb{P}$ admits a forcing relation $\Vdash_{\mathbb{P}}$, that is, a relation satisfying the forcing relation recursion -- it follows that statements true in the…
We present a logic for the reasoning about necessity and justifications which is independent from relational semantics. We choose the concept of justification -- coming from a class of "Justification Logics" (Artemov 2008, Fitting 2009) --…
Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete Boolean algebra in $M.$ (In general a proper class.) We define a generalized notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing extension of…
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 phi holding in some forcing extension V^P and all subsequent extensions V^P*Q holds…
The primary purpose of this article is to show that a certain natural set of axioms yields a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely)…
We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A…
In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(\omega_3,\omega_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\omega_2$ and $\omega_3$. In this paper we formulate a…
We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner…
The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $\sigma$-continuity of a certain Borel not $\sigma$-continuous function. We give a characterization of this forcing in the language of trees and using this…
Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add…
The purpose of this article is to give a presentation of the method of forcing aimed at someone with a minimal knowledge of set theory and logic. The emphasis will be on how the method can be used to prove theorems in ZFC.
A famous result, conjectured by G\"odel in 1932 and proved by McKinsey and Tarski in 1948, says that $\varphi$ is a theorem of intuitionistic propositional logic IPC iff its G\"odel-translation $\varphi'$ is a theorem of modal logic S4. In…