Related papers: A simple maximality principle
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 it is true in all forcing extensions of V.…
The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in…
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…
It is shown that the boldface maximality principle for subcomplete forcing, together with the assumption that the universe has only set-many grounds, implies the existence of a (parameter-free) definable well-ordering of…
In this paper we prove that the maximum principle in forcing is equivalent to the axiom of choice. The maximum principle is the property of forcing: p ||- exists x theta(x) iff for some name tau p ||- theta(tau). We also look at three…
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 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…
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…
In this paper we consider the Foreman's maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We prove that it is…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…
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…
We consider the modality "$\varphi$ is true in every $\sigma$-centered forcing extension", denoted $\square\varphi$, and its dual "$\varphi$ is true in some $\sigma$-centered forcing extension", denoted $\lozenge\varphi$ (where $\varphi$ is…
Given a Woodin cardinal $\delta$, I show that if $F$ is any Easton function with $F"\delta\subseteq\delta$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^\gamma= F(\gamma)$ for each regular cardinal…
We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we…
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…
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…
We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…
We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $\Sigma_n$-$\mathsf{MP}(\mathbb R)$ and $\Pi_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle…
We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration theorem that applies to many classical forcings…
The Recurrence Axiom for a class $\mathcal{P}$ of \pos\ and a set $A$ of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from $A$ is forced by a poset in $\mathcal{P}$, then there is a…