Related papers: Infinite forcing and the generic multiverse
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a…
We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which…
We analyse the Boolean-valued random forcing $B_{M,\Omega}$ in bounded arithmetics developed in Krajicek (Forcing with random variables and proof complexity, vol. 382, Cambridge University Press, 2011) from the perspective of the forcing in…
A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of $V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method…
We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are…
We make use of a finite support product of $\omega_1$ clones of the Jensen minimal $\varPi^1_2$ singleton forcing to obtain a model of ZFC in which every non-empty lightface analytically definable set of reals contains a lightface…
In the present paper, classical tools of convex analysis are used to study the solution set to a certain class of set-inclusive generalized equations. A condition for the solution existence and global error bounds is established, in the…
Foundation models contain a wealth of information from their vast number of training samples. However, most prior arts fail to extract this information in a precise and efficient way for small sample sizes. In this work, we propose a…
Although inverse limits with factor spaces indexed by the positive integers are most commonly studied, Ingram and Mahavier have defined inverse limits with set-valued functions broadly enough for any directed index set to be used. In this…
Partial descriptions of the Universe are presented in the form of linear equations considered in the free (full, super) Fock space. The universal properties of these equations are discussed. The closure problem caused by computational and…
In the sequential decision making setting, an agent aims to achieve systematic generalization over a large, possibly infinite, set of environments. Such environments are modeled as discrete Markov decision processes with both states and…
The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts.
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…
In this paper we investigate to which extent a very simple and natural "reachability as deducibility" approach, originated in the research in formal methods in security, is applicable to the automated verification of large classes of…
We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously…
We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Lof's type theory (hence…
We study models M of set theory that are "condensable", in the sense that there is an "ordinal" v of M such that the rank initial segment of M determined by v is both isomorphic to M, and also an elementary submodel of M for infinitary…
A growing number of empirical models exhibit set-valued predictions. This paper develops a tractable inference method with finite-sample validity for such models. The proposed procedure uses a robust version of the universal inference…
We prove that Solovay's set $\Sigma$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation.