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Herbrandized modified realizability 修正的可实现性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-04-04 DOI: 10.1007/s00153-024-00917-6
Gilda Ferreira, Paulo Firmino
{"title":"Herbrandized modified realizability","authors":"Gilda Ferreira,&nbsp;Paulo Firmino","doi":"10.1007/s00153-024-00917-6","DOIUrl":"10.1007/s00153-024-00917-6","url":null,"abstract":"<div><p>Realizability notions in mathematical logic have a long history, which can be traced back to the work of Stephen Kleene in the 1940s, aimed at exploring the foundations of intuitionistic logic. Kleene’s initial realizability laid the ground for more sophisticated notions such as Kreisel’s modified realizability and various modern approaches. In this context, our work aligns with the lineage of realizability strategies that emphasize the accumulation, rather than the propagation of precise witnesses. In this paper, we introduce a new notion of realizability, namely <i>herbrandized modified realizability</i>. This novel form of (cumulative) realizability, presented within the framework of semi-intuitionistic logic is based on a recently developed <i>star combinatory calculus</i>, which enables the gathering of witnesses into nonempty finite sets. We also show that the previous analysis can be extended from logic to (Heyting) arithmetic.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"703 - 721"},"PeriodicalIF":0.3,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00917-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Indiscernibles and satisfaction classes in arithmetic 算术中的不可分性和满足类
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-03-28 DOI: 10.1007/s00153-024-00915-8
Ali Enayat
{"title":"Indiscernibles and satisfaction classes in arithmetic","authors":"Ali Enayat","doi":"10.1007/s00153-024-00915-8","DOIUrl":"10.1007/s00153-024-00915-8","url":null,"abstract":"<div><p>We investigate the theory Peano Arithmetic with Indiscernibles (<span>(textrm{PAI})</span>). Models of <span>(textrm{PAI})</span> are of the form <span>(({mathcal {M}},I))</span>, where <span>({mathcal {M}})</span> is a model of <span>(textrm{PA})</span>, <i>I</i> is an unbounded set of order indiscernibles over <span>({mathcal {M}})</span>, and <span>(({mathcal {M}},I))</span> satisfies the extended induction scheme for formulae mentioning <i>I</i>. Our main results are Theorems A and B following. <b>Theorem A.</b> <i>Let</i> <span>({mathcal {M}})</span> <i>be a nonstandard model of</i> <span>(textrm{PA})</span><i> of any cardinality</i>. <span>(mathcal {M })</span> <i>has an expansion to a model of </i><span>(textrm{PAI})</span> <i>iff</i> <span>( {mathcal {M}})</span> <i>has an inductive partial satisfaction class.</i> Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of <span>(textrm{PA})</span>: <b>Corollary.</b> <i>A countable model</i> <span>({mathcal {M}})</span> of <span>(textrm{PA})</span> <i>is recursively saturated iff </i><span>({mathcal {M}})</span> <i>has an expansion to a model of </i><span>(textrm{PAI})</span>. <b>Theorem B.</b> <i>There is a sentence </i><span>(alpha )</span> <i> in the language obtained by adding a unary predicate</i> <i>I</i>(<i>x</i>) <i>to the language of arithmetic such that given any nonstandard model </i><span>({mathcal {M}})</span> <i>of</i> <span>(textrm{PA})</span><i> of any cardinality</i>, <span>({mathcal {M}})</span> <i>has an expansion to a model of </i><span>(text {PAI}+alpha )</span> <i>iff</i> <span>({mathcal {M}})</span> <i>has a inductive full satisfaction class.</i></p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"655 - 677"},"PeriodicalIF":0.3,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00915-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cohesive powers of structures 结构的凝聚力
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-03-28 DOI: 10.1007/s00153-024-00916-7
Valentina Harizanov, Keshav Srinivasan
{"title":"Cohesive powers of structures","authors":"Valentina Harizanov,&nbsp;Keshav Srinivasan","doi":"10.1007/s00153-024-00916-7","DOIUrl":"10.1007/s00153-024-00916-7","url":null,"abstract":"<div><p>A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its effective power over a cohesive set of natural numbers. A cohesive set is an infinite set of natural numbers that is indecomposable with respect to computably enumerable sets. It plays the role of an ultrafilter, and the elements of a cohesive power are the equivalence classes of certain partial computable functions determined by the cohesive set. Thus, unlike many classical ultrapowers, a cohesive power is a countable structure. In this paper we focus on cohesive powers of graphs, equivalence structures, and computable structures with a single unary function satisfying various properties, which can also be viewed as directed graphs. For these computable structures, we investigate the isomorphism types of their cohesive powers, as well as the properties of cohesive powers when they are not isomorphic to the original structure.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"679 - 702"},"PeriodicalIF":0.3,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140324603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Pcf without choice Sh835 Pcf 无选择 Sh835
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-03-22 DOI: 10.1007/s00153-023-00900-7
Saharon Shelah
{"title":"Pcf without choice Sh835","authors":"Saharon Shelah","doi":"10.1007/s00153-023-00900-7","DOIUrl":"10.1007/s00153-023-00900-7","url":null,"abstract":"<div><p>We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of <span>(lambda )</span> is well ordered for every <span>(lambda )</span> (really local version for a given <span>(lambda )</span>). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable and we can prove cardinal inequalities. Solving some open problems, we prove that if <span>(mu&gt; kappa = textrm{cf}(mu ) &gt; aleph _{0},)</span> then from a well ordering of <span>({mathscr {P}}({mathscr {P}}(kappa )) cup {}^{kappa &gt;} mu )</span> we can define a well ordering of <span>({}^{kappa } mu .)</span></p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"623 - 654"},"PeriodicalIF":0.3,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140202383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On computable numberings of families of Turing degrees 论图灵度族的可计算编号
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-03-18 DOI: 10.1007/s00153-024-00914-9
Marat Faizrahmanov
{"title":"On computable numberings of families of Turing degrees","authors":"Marat Faizrahmanov","doi":"10.1007/s00153-024-00914-9","DOIUrl":"10.1007/s00153-024-00914-9","url":null,"abstract":"<div><p>In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each of its elements computable numbering even if it has no principal numberings. It follows from results by Mal’tsev and Ershov that complete numberings have nice programming tools and computational properties such as Kleene’s recursion theorems, Rice’s theorem, Visser’s ADN theorem, etc. Thus, every computable family of Turing degrees has a computable numbering with these properties. Finally, we prove that the Rogers semilattice of each such non-empty non-singleton family is infinite and is not a lattice.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"609 - 622"},"PeriodicalIF":0.3,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140171806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Around accumulation points and maximal sequences of indiscernibles 堆积点周围和不可辨别的最大序列
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-03-13 DOI: 10.1007/s00153-024-00913-w
Moti Gitik
{"title":"Around accumulation points and maximal sequences of indiscernibles","authors":"Moti Gitik","doi":"10.1007/s00153-024-00913-w","DOIUrl":"10.1007/s00153-024-00913-w","url":null,"abstract":"<div><p>Answering a question of Mitchell (Trans Am Math Soc 329(2):507–530, 1992) we show that a limit of accumulation points can be singular in <span>({mathcal {K}})</span>. Some additional constructions are presented.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"591 - 608"},"PeriodicalIF":0.3,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00913-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140129066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Varieties of truth definitions 真理定义的多样性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-03-09 DOI: 10.1007/s00153-024-00909-6
Piotr Gruza, Mateusz Łełyk
{"title":"Varieties of truth definitions","authors":"Piotr Gruza,&nbsp;Mateusz Łełyk","doi":"10.1007/s00153-024-00909-6","DOIUrl":"10.1007/s00153-024-00909-6","url":null,"abstract":"<div><p>We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence <span>(alpha )</span> which extends a weak arithmetical theory (which we take to be <span>({{,mathrm{IDelta _{0}+exp },}})</span>) such that for some formula <span>(Theta )</span> and any arithmetical sentence <span>(varphi )</span>, <span>(Theta (ulcorner varphi urcorner )equiv varphi )</span> is provable in <span>(alpha )</span>. We say that a sentence <span>(beta )</span> is definable in a sentence <span>(alpha )</span>, if there exists an unrelativized translation from the language of <span>(beta )</span> to the language of <span>(alpha )</span> which is identity on the arithmetical symbols and such that the translation of <span>(beta )</span> is provable in <span>(alpha )</span>. Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not <span>(Sigma _2)</span>-definable in the standard model of arithmetic. We conclude by remarking that no <span>(Sigma _2)</span>-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"563 - 589"},"PeriodicalIF":0.3,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140099344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Essential hereditary undecidability 基本遗传不可判定性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-03-01 DOI: 10.1007/s00153-024-00911-y
Albert Visser
{"title":"Essential hereditary undecidability","authors":"Albert Visser","doi":"10.1007/s00153-024-00911-y","DOIUrl":"10.1007/s00153-024-00911-y","url":null,"abstract":"<div><p>In this paper we study <i>essential hereditary undecidability</i>. Theories with this property are a convenient tool to prove undecidability of other theories. The paper develops the basic facts concerning essentially hereditary undecidability and provides salient examples, like a construction of essentially hereditarily undecidable theories due to Hanf and an example of a rather natural essentially hereditarily undecidable theory strictly below <span>R</span>. We discuss the (non-)interaction of essential hereditary undecidability with recursive boolean isomorphism. We develop a reduction relation <i>essential tolerance</i>, or, in the converse direction, <i>lax interpretability</i> that interacts in a good way with essential hereditary undecidability. We introduce the class of <span>(Sigma ^0_1)</span>-friendly theories and show that <span>(Sigma ^0_1)</span>-friendliness is sufficient but not necessary for essential hereditary undecidability. Finally, we adapt an argument due to Pakhomov, Murwanashyaka and Visser to show that there is no interpretability minimal essentially hereditarily undecidable theory.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"529 - 562"},"PeriodicalIF":0.3,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00911-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140017680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the extendability to (mathbf {Pi }_3^0) ideals and Katětov order 关于向 $$mathbf {Pi }_3^0$$ 理想和 Katětov 秩的可扩展性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-02-26 DOI: 10.1007/s00153-024-00912-x
Jialiang He, Jintao Luo, Shuguo Zhang
{"title":"On the extendability to (mathbf {Pi }_3^0) ideals and Katětov order","authors":"Jialiang He,&nbsp;Jintao Luo,&nbsp;Shuguo Zhang","doi":"10.1007/s00153-024-00912-x","DOIUrl":"10.1007/s00153-024-00912-x","url":null,"abstract":"<div><p>We show that there is a <span>( varvec{Sigma }_4^0)</span> ideal such that it’s neither extendable to any <span>( varvec{Pi }_3^0)</span> ideal nor above the ideal <span>( textrm{Fin}times textrm{Fin} )</span> in the sense of Katětov order, answering a question from M. Hrušák.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"523 - 528"},"PeriodicalIF":0.3,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Errata: on the role of the continuum hypothesis in forcing principles for subcomplete forcing 勘误：关于连续体假设在次完全强迫原理中的作用
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-02-19 DOI: 10.1007/s00153-024-00905-w
Gunter Fuchs
{"title":"Errata: on the role of the continuum hypothesis in forcing principles for subcomplete forcing","authors":"Gunter Fuchs","doi":"10.1007/s00153-024-00905-w","DOIUrl":"10.1007/s00153-024-00905-w","url":null,"abstract":"<div><p>In this note, I will list instances where in the literature on subcomplete forcing and its forcing principles (mostly in articles of my own), the assumption of the continuum hypothesis, or that we are working above the continuum, was omitted. I state the correct statements and provide or point to correct proofs. There are also some new results, most of which revolve around showing the necessity of the extra assumption.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 5-6","pages":"509 - 521"},"PeriodicalIF":0.3,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139910639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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