arXiv 0807.12542008-07-08math.LO

$σ$-continuity and related forcings

Marcin Sabok

英文摘要：

The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $σ$-continuity of a certain Borel not $σ$-continuous function. We give a characterization of this forcing in the language of trees and using this characterization we establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show it is not the case with Steprans forcing. We also establish a connection between Steprans forcing and Miller forcing thus giving a new description of the latter. Eventually, we exhibit a variety of forcing notions which do not have continuous reading of names in any presentation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Modern Forcing Techniques related to Finite Support Iteration: Ultrapowers, templates, and submodels

This is an expository paper about several sophisticated forcing techniques closely related to standard finite support iterations of ccc partial orders. We focus on the four topics of ultrapowers of forcing notions, iterations along templates, Boolean ultrapowers of forcing notions, and restrictions of forcing notions to elementary submodels.

2022-02-02math.LO

The modal logic of $σ$-centered forcing and related forcing classes

We consider the modality "$φ$ is true in every $σ$-centered forcing extension", denoted $\squareφ$, and its dual "$φ$ is true in some $σ$-centered forcing extension", denoted $\lozengeφ$ (where $φ$ is a statement in set theory), which give rise to the notion of a "principle of $σ$-centered forcing". We prove that if ZFC is consistent, then the modal logic of $σ$-centered forcing, i.e. the ZFC-provable principles of $σ$-centered forcing, is exactly $\mathsf{S4.2}$. We also generalize this result to other related classes of forcing.

2019-12-11math.LO

From A1 to D5: Towards a Forcing-Related Classification of Relational Structures

We investigate the partial orderings of the form (P(X),\subset), where X is a relational structure and P(X) the set of the domains of its isomorphic substructures. A rough classification of countable binary structures corresponding to the forcing-related properties of the posets of their copies is obtained.

2013-03-11math.LO

力迫公理、可逼近性与平稳集反射

Forcing axioms, approachability, and stationary set reflection

本文证明平稳集反射与内部可逼近性的多种定理，指出Fuchino-Usuba蕴含不可逆，加强并简化Krueger结果，并借鉴Woodin思想统一力迫公理保持论证。

2020-01-09math.LO

Some results related to the continuity problem

The continuity problem, i.e., the question whether effective maps between effectively given topological spaces are effectively continuous, is reconsidered. In earlier work it was shown that this is always the case, if the effective map also has a witness for noninclusion. The extra condition does not have an obvious topological interpretation. As is shown in the present paper, it appears naturally where in the classical proof that sequentially continuous maps are continuous the Axiom of Choice is used. The question is therefore whether the witness condition appears in the general continuity th…

2016-03-01math.LO

Strongly proper forcing and some problems of Foreman

We provide solutions to several problems of Foreman about ideals, several of which are closely related to Mitchell's notion of \emph{strongly proper} forcing. We prove: 1) Presaturation of a normal ideal implies projective antichain catching, enabling us to provide a solution to a problem from Foreman~\cite{MR2768692} about ideal projections which is more comprehensive and simpler than the solution obtained in \cite{MR3343538}. 2) We solve an older question from Foreman~\cite{MR819932} about the relationship between generic hugeness and generic almost hugeness. 3) Finally, we provide solutions…

2018-03-12math.LO

更多相关论文收起更多论文

Reverse Mathematics of Brouwer's continuity theorem and related principles

In intuitionistic mathematics, the Brouwer Continuity Theorem states that all total real functions are (uniformly) continuous on the unit interval. We study this theorem and related principles from the point of view of Reverse Mathematics over a base theory accommodating higher types and Nonstandard Analysis. With regard to the bigger picture, Reverse Mathematics provides a classification of theorems of ordinary mathematics based on computability. Our aim is to provide an alternative classification of theorems based on the central tenet of Feferman's Explicit Mathematics}, namely that a proof …

2015-02-12math.LO

Forcing with Adequate Sets of Models as Side Conditions

We present a general framework for forcing on $ω_2$ with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on $ω_2$, adding a nonreflecting stationary subset of $ω_2 \cap \textrm{cof}(ω)$, and adding an $ω_1$-Kurepa tree.

2016-06-09math.LO

Cofinalities of Marczewski-like ideals

We show that the cofinalities of both the Miller ideal m^0 (the sigma-ideal naturally related to Miller forcing) and the Laver ideal ell^0 (related to Laver forcing) are larger than the size of the continuum in ZFC.

2016-11-24math.LO

Selective Independence and $h$-Perfect Tree Forcing Notions

Generalizing the proof for Sacks forcing, we show that the $h$-perfect tree forcing notions introduced by Goldstern, Judah and Shelah preserve selective independent families even when iterated. As a result we obtain new proofs of the consistency of $\mathfrak{i} = \mathfrak{u} < \mathrm{non} (\mathcal N) = \mathrm{cof} (\mathcal N)$ and $\mathfrak{i} < \mathfrak{u} = \mathrm{non} (\mathcal N) = \mathrm{cof}( \mathcal N)$ as well as some related results.

2022-02-24math.LO

Proper forcing and rectangular Ramsey theorems

I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.

2003-11-09math.COmath.LO

Combinatorial properties of classical forcing notions

We discuss the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum (like the unbounding or the dominating number or the cardinals related to measure and category on the real line). For random and Cohen forcing, this question was investigated by Cicho'n and Pawlikowski; for Hechler forcing, by Judah, Shelah and myself. We show here: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size omega_1; (2) Laver and Mathias forcing collapse the dominating number …

1993-09-09math.LO

The Diagonal Strong Reflection Principle and its fragments

A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated to arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle are analyzed, and consequences are presented. Some of these consequences are ``exact'' versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

2021-08-10math.LO

Definable maximal discrete sets in forcing extensions

Let $\mathcal R$ be a $Σ^1_1$ binary relation, and recall that a set $A$ is $\mathcal R$-discrete if no two elements of $A$ are related by $\mathcal R$. We show that in the Sacks and Miller forcing extensions of $L$ there is a $Δ^1_2$ maximal $\mathcal{R}$-discrete set. We use this to answer in the negative the main question posed in \cite{Fischer2010} by showing that in the Sacks and Miller extensions there is a $Π^1_1$ maximal orthogonal family ("mof") of Borel probability measures on Cantor space. By contrast, we show that if there is a Mathias real over $L$ then there are no $Σ^1_2$ mofs.

2022-09-29math.LO

Sweet & Sour and other flavours of ccc forcing notions

The present paper has three themes. First, we continue the investigations started in Judah, Roslanowski and Shelah \math.LO/9310224 and Roslanowski and Shelah math.LO/9807172, math.LO/9703222, and we investigate the method of norms on possibilities in the context of ccc forcing notions, getting a number of constructions of nicely definable ccc forcings. The second theme of the paper is a part of the general program ``how special are random and Cohen forcing notions (or: the respective ideals)''. Shelah math.LO/9303208 shows that the two forcing notions may occupy special positions in the realm…

2001-12-20math.LO

A parallel to the null ideal for inaccessible lambda. Part I

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 forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…

2017-01-18math.LO

通用支配与理想化力迫

Universal domination and idealized forcing

University of Vienna（维也纳大学）

该研究引入波兰空间可定义σ-理想的通用性质，将其作为理想化力迫固有性的基准并统一相关结果，借此回答了两个关于理想化力迫的问题。

2026-08-20math.LO