Choice principles in local mantles

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly Pub Date : 2022-05-07 DOI:10.1002/malq.202000089

Farmer Schlutzenberg

{"title":"Choice principles in local mantles","authors":"Farmer Schlutzenberg","doi":"10.1002/malq.202000089","DOIUrl":null,"url":null,"abstract":"Assume <math>\n <semantics>\n <mi>ZFC</mi>\n <annotation>$\\mathsf {ZFC}$</annotation>\n </semantics></math>. Let κ be a cardinal. A <math>\n <semantics>\n <mrow>\n <mo><</mo>\n <mspace></mspace>\n <mi>κ</mi>\n </mrow>\n <annotation>${\\mathord {<}\\hspace{1.111pt}\\kappa }$</annotation>\n </semantics></math>-ground\nis a transitive proper class W modelling <math>\n <semantics>\n <mi>ZFC</mi>\n <annotation>$\\mathsf {ZFC}$</annotation>\n </semantics></math> such that V is a generic extension of W via a forcing <math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mo>∈</mo>\n <mi>W</mi>\n </mrow>\n <annotation>$\\mathbb {P}\\in W$</annotation>\n </semantics></math> of cardinality <math>\n <semantics>\n <mrow>\n <mo><</mo>\n <mspace></mspace>\n <mi>κ</mi>\n </mrow>\n <annotation>${\\mathord {<}\\hspace{1.111pt}\\kappa }$</annotation>\n </semantics></math>. The κ-mantle <math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mi>κ</mi>\n </msub>\n <annotation>$\\mathcal {M}_\\kappa$</annotation>\n </semantics></math> is the intersection of all <math>\n <semantics>\n <mrow>\n <mo><</mo>\n <mspace></mspace>\n <mi>κ</mi>\n </mrow>\n <annotation>${\\mathord {<}\\hspace{1.111pt}\\kappa }$</annotation>\n </semantics></math>-grounds. We prove that certain partial choice principles in <math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mi>κ</mi>\n </msub>\n <annotation>$\\mathcal {M}_\\kappa$</annotation>\n </semantics></math> are the consequence of κ being inaccessible/weakly compact, and some other related facts.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 3","pages":"264-278"},"PeriodicalIF":0.4000,"publicationDate":"2022-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202000089","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202000089","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}

引用次数: 2

Abstract

Assume $ZFC$ . Let κ be a cardinal. A $< κ$ -ground is a transitive proper class W modelling $ZFC$ such that V is a generic extension of W via a forcing $P \in W$ of cardinality $< κ$ . The κ-mantle $M_{κ}$ is the intersection of all $< κ$ -grounds. We prove that certain partial choice principles in $M_{κ}$ are the consequence of κ being inaccessible/weakly compact, and some other related facts.

查看原文本刊更多论文

当地的选择原则

假设ZFC $\mathsf {ZFC}$。设κ为基数。一个 & lt;κ ${\mathord {<}\hspace{1.111pt}\kappa}$ - ground是一个可传递的固有类W，它对ZFC $\mathsf {ZFC}$建模，使得V是W的一个泛型扩展，通过在W$的基数中强制P∈W$ \mathbb {P}\& lt;κ ${\mathord {<}\hspace{1.111pt}\kappa}$。κ-地幔M κ $\mathcal {M}_\kappa$是所有<κ ${\mathord {<}\hspace{1.111pt}\kappa}$ -grounds。我们证明了M κ $\mathcal {M}_\kappa$中的某些部分选择原理是κ不可及/弱紧致的结果，以及其他一些相关事实。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematical Logic Quarterly 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.