Related papers: A note on the modal logic of symmetric extensions
We give a brief account of 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 technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique…
The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver's theorem and Bukovsk\'y's theorem assert that set-generic extensions of a given…
We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present…
Recent work in set theory indicates that there are many different notions of 'set', each captured by a different collection of axioms, as proposed by J. Hamkins in [Ham11]. In this paper we strive to give one class theory that allows for a…
The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of ZF between the ground model and the generic extension, and often the axiom of choice…
Asymmetric combination of logics is a formal process that develops the characteristic features of a specific logic on top of another one. Typical examples include the development of temporal, hybrid, and probabilistic dimensions over a…
Propositional term modal logic is interpreted over Kripke structures with unboundedly many accessibility relations and hence the syntax admits variables indexing modalities and quantification over them. This logic is undecidable, and we…
Relation-changing modal logics are extensions of the basic modal logic that allow changes to the accessibility relation of a model during the evaluation of a formula. In particular, they are equipped with dynamic modalities that are able to…
We bring forward a logical system of transition algebras that enhances many-sorted first-order logic using features from dynamic logics. The sentences we consider include compositions, unions, and transitive closures of transition…
This paper investigates the extension of lattice-based logics into modal languages. We observe that such extensions admit multiple approaches, as the interpretation of the necessity operator is not uniquely determined by the underlying…
An introduction is given to the logic of sheaves of structures and to set theoretic forcing constructions based on this logic. Using these tools, it is presented an alternative proof of the independence of the Continuum Hypothesis; which…
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…
We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and L\"owe, including…
We prove some general theorems for preserving Dependent Choice when taking symmetric extensions, some of which are unwritten folklore results. We apply these to various constructions to obtain various simple consistency proofs.
We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial non-deterministic…
Proof assistants play a dual role as programming languages and logical systems. As programming languages, proof assistants offer standard modularity mechanisms such as first-class functions, type polymorphism and modules. As logical…
Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic…
It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models $V \subseteq…
This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized…