相关论文: Non-elementary proper forcing notions
We make use of a finite support product of Jensen forcing to define a model in which there is a countable non-empty lightface $\Pi^1_2$ set of reals containing no ordinal-definable real.
We give a self-contained proof of the preservation theorem for proper countable support iterations known as "tools-preservation," "Case A" or "first preservation theorem" in the literature. We do not assume that the forcings add reals.
In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was…
Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) F{\o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of $\{0,1\}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = \{1,2,...,n\}$, the…
We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an…
This paper presents a type theory with a form of equality reflection: provable equalities can be used to coerce the type of a term. Coercions and other annotations, including implicit arguments, are dropped during reduction of terms. We…
Perhaps the most prominent current definition of (actual) causality is due to Halpern and Pearl. It is defined using causal models (also known as structural equations models). We abstract the definition, extracting its key features, so that…
We give category-theoretic reformulations of stability, NIP, NTP, and non-dividing by observing that their characterisations in terms of indiscernible sequences are naturally expressed as Quillen lifting properties %(negation) of certain…
By a virtual model, we mean a model of set theory which is elementary in its transitive closure. Virtual models are first used by Neeman \cite{neeman2014forcing} to iterate forcing. That paper is concerned with proper forcing. The method…
The work demonstrates that brain might reflect the external world causal relationships in the form of a logically consistent and prognostic model of reality, which shows up as consciousness. The paper analyses and solves the problem of…
We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the…
We present a family of paraconsistent counterparts of the constructive modal logic CK. These logics aim to formalise reasoning about contradictory but non-trivial propositional attitudes like beliefs or obligations. We define their…
We study the spectrum of forcing notions between the iterations of $\sigma$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals as well as…
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the…
We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a…
This paper investigates situations where a property of a ring can be tested on a set of "prime right ideals." Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff…
We present the classical theory of preservation of $\sqsubset$-unbounded families in generic extensions by ccc posets, where $\sqsubset$ is a definable relation of certain type on spaces of real numbers, typically associated with some…
We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of…
For models of concurrent and distributed systems, it is important and also challenging to establish correctness in terms of safety and/or liveness properties. Theories of distributed systems consider equivalences fundamental, since they (1)…
Recent results show that a constraint satisfaction problem (CSP) defined over rational numbers with their natural ordering has a solution if and only if it has a definable solution. The proof uses advanced results from topology and modern…