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Definable Tietze extension property in o-minimal expansions of ordered groups 有序群o-极小展开中可定义的Tietze扩张性质
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-15 DOI: 10.1007/s00153-023-00875-5
Masato Fujita
{"title":"Definable Tietze extension property in o-minimal expansions of ordered groups","authors":"Masato Fujita","doi":"10.1007/s00153-023-00875-5","DOIUrl":"10.1007/s00153-023-00875-5","url":null,"abstract":"<div><p>The following two assertions are equivalent for an o-minimal expansion of an ordered group <span>(mathcal M=(M,&lt;,+,0,ldots ))</span>. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function <span>(f:A rightarrow M)</span> defined on a definable closed subset of <span>(M^n)</span> has a definable continuous extension <span>(F:M^n rightarrow M)</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44364332","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
Correction to: Towers, mad families, and unboundedness 更正:塔，疯狂的家庭，和无边界
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-04-03 DOI: 10.1007/s00153-023-00870-w
Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky
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引用次数: 0
Correction to: Towers, mad families, and unboundedness 更正:塔，疯狂的家庭，和无边界
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-04-03 DOI: 10.1007/s00153-023-00870-w
Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky
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引用次数: 0
Consistency and interpolation in linear continuous logic 线性连续逻辑中的一致性与插值
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-03-21 DOI: 10.1007/s00153-023-00869-3
Mahya Malekghasemi, Seyed-Mohammad Bagheri
{"title":"Consistency and interpolation in linear continuous logic","authors":"Mahya Malekghasemi,&nbsp;Seyed-Mohammad Bagheri","doi":"10.1007/s00153-023-00869-3","DOIUrl":"10.1007/s00153-023-00869-3","url":null,"abstract":"<div><p>We prove Robinson consistency theorem as well as Craig, Lyndon and Herbrand interpolation theorems in linear continuous logic.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43271849","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
The axiom of choice in metric measure spaces and maximal (delta )-separated sets 度量度量空间和极大(delta )分离集中的选择公理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-03-13 DOI: 10.1007/s00153-023-00868-4
Michał Dybowski, Przemysław Górka
{"title":"The axiom of choice in metric measure spaces and maximal (delta )-separated sets","authors":"Michał Dybowski,&nbsp;Przemysław Górka","doi":"10.1007/s00153-023-00868-4","DOIUrl":"10.1007/s00153-023-00868-4","url":null,"abstract":"<div><p>We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal <span>(delta )</span>-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00868-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50048305","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
A topological completeness theorem for transfinite provability logic 超限可证明性逻辑的拓扑完备性定理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-02-22 DOI: 10.1007/s00153-023-00863-9
Juan P. Aguilera
{"title":"A topological completeness theorem for transfinite provability logic","authors":"Juan P. Aguilera","doi":"10.1007/s00153-023-00863-9","DOIUrl":"10.1007/s00153-023-00863-9","url":null,"abstract":"<div><p>We prove a topological completeness theorem for the modal logic <span>(textsf{GLP})</span> containing operators <span>({langle xi rangle :xi in textsf{Ord}})</span> intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space <i>X</i>, any sentence <span>(phi )</span> consistent with <span>(textsf{GLP})</span> can be satisfied on a polytopological space based on finitely many Icard topologies constructed over <i>X</i> and corresponding to the finitely many modalities that occur in <span>(phi )</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00863-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50042104","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: 1
Sets of real numbers closed under Turing equivalence: applications to fields, orders and automorphisms 在图灵等价下闭合的实数集：域、阶和自同构的应用
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-02-16 DOI: 10.1007/s00153-023-00865-7
Iván Ongay-Valverde
{"title":"Sets of real numbers closed under Turing equivalence: applications to fields, orders and automorphisms","authors":"Iván Ongay-Valverde","doi":"10.1007/s00153-023-00865-7","DOIUrl":"10.1007/s00153-023-00865-7","url":null,"abstract":"<div><p>In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44115779","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
Ideals with Smital properties 具有Smital属性的理想
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-02-14 DOI: 10.1007/s00153-023-00867-5
Marcin Michalski, Robert Rałowski, Szymon Żeberski
{"title":"Ideals with Smital properties","authors":"Marcin Michalski,&nbsp;Robert Rałowski,&nbsp;Szymon Żeberski","doi":"10.1007/s00153-023-00867-5","DOIUrl":"10.1007/s00153-023-00867-5","url":null,"abstract":"<div><p>A <span>(sigma )</span>-ideal <span>(mathcal {I})</span> on a Polish group <span>((X,+))</span> has the Smital Property if for every dense set <i>D</i> and a Borel <span>(mathcal {I})</span>-positive set <i>B</i> the algebraic sum <span>(D+B)</span> is a complement of a set from <span>(mathcal {I})</span>. We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are <span>(mathfrak {c})</span> many maximal invariant <span>(sigma )</span>-ideals with Borel bases on the Cantor space <span>(2^omega )</span>.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00867-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45603623","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
Towers, mad families, and unboundedness 塔楼、疯狂的家庭和无边界。
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-02-12 DOI: 10.1007/s00153-023-00861-x
Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky
{"title":"Towers, mad families, and unboundedness","authors":"Vera Fischer,&nbsp;Marlene Koelbing,&nbsp;Wolfgang Wohofsky","doi":"10.1007/s00153-023-00861-x","DOIUrl":"10.1007/s00153-023-00861-x","url":null,"abstract":"<div><p>We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are <span>({mathcal {B}})</span>-Canjar for any countably directed unbounded family <span>({mathcal {B}})</span> of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that <span>({mathfrak {b}}=omega _1)</span> in every extension by the above forcing notions.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00861-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9566941","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: 2
Involutive symmetric Gödel spaces, their algebraic duals and logic 对合对称Gödel空间及其代数对偶与逻辑
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-02-10 DOI: 10.1007/s00153-023-00866-6
A. Di Nola, R. Grigolia, G. Vitale
{"title":"Involutive symmetric Gödel spaces, their algebraic duals and logic","authors":"A. Di Nola,&nbsp;R. Grigolia,&nbsp;G. Vitale","doi":"10.1007/s00153-023-00866-6","DOIUrl":"10.1007/s00153-023-00866-6","url":null,"abstract":"<div><p>It is introduced a new algebra <span>((A, otimes , oplus , *, rightharpoonup , 0, 1))</span> called <span>(L_PG)</span>-algebra if <span>((A, otimes , oplus , *, 0, 1))</span> is <span>(L_P)</span>-algebra (i.e. an algebra from the variety generated by perfect <i>MV</i>-algebras) and <span>((A,rightharpoonup , 0, 1))</span> is a Gödel algebra (i.e. Heyting algebra satisfying the identity <span>((x rightharpoonup y ) vee (y rightharpoonup x ) =1))</span>. The lattice of congruences of an <span>(L_PG)</span> -algebra <span>((A, otimes , oplus , *, rightharpoonup , 0, 1))</span> is isomorphic to the lattice of Skolem filters (i.e. special type of <i>MV</i>-filters) of the <i>MV</i>-algebra <span>((A, otimes , oplus , *, 0, 1))</span>. The variety <span>(mathbf {L_PG})</span> of <span>(L_PG)</span> -algebras is generated by the algebras <span>((C, otimes , oplus , *, rightharpoonup , 0, 1))</span> where <span>((C, otimes , oplus , *, 0, 1))</span> is Chang <i>MV</i>-algebra. Any <span>(L_PG)</span> -algebra is bi-Heyting algebra. The set of theorems of the logic <span>(L_PG)</span> is recursively enumerable. Moreover, we describe finitely generated free <span>(L_PG)</span>-algebras.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00866-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46294733","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
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