Archive for Mathematical Logic最新文献

{"title":"On absorption’s formula definable semigroups of complete theories","authors":"Mahsut Bekenov,&nbsp;Aida Kassatova,&nbsp;Anvar Nurakunov","doi":"10.1007/s00153-024-00937-2","DOIUrl":"10.1007/s00153-024-00937-2","url":null,"abstract":"<div><p>On the set of all first-order complete theories <span>(T(sigma ))</span> of a language <span>(sigma )</span> we define a binary operation <span>({cdot })</span> by the rule: <span>(Tcdot S= {{,textrm{Th},}}({Atimes Bmid Amodels T ,,text {and},, Bmodels S}))</span> for any complete theories <span>(T, Sin T(sigma ))</span>. The structure <span>(langle T(sigma );cdot rangle )</span> forms a commutative semigroup. A subsemigroup <i>S</i> of <span>(langle T(sigma );cdot rangle )</span> is called an <i>absorption’s formula definable semigroup</i> if there is a complete theory <span>(Tin T(sigma ))</span> such that <span>(S=langle {Xin T(sigma )mid Xcdot T=T};cdot rangle )</span>. In this event we say that a theory <i>T</i> <i>absorbs</i> <i>S</i>. In the article we show that for any absorption’s formula definable semigroup <i>S</i> the class <span>({{,textrm{Mod},}}(S)={Ain {{,textrm{Mod},}}(sigma )mid Amodels T_0,,text {for some},, T_0in S})</span> is axiomatizable, and there is an idempotent element <span>(Tin S)</span> that absorbs <i>S</i>. Moreover, <span>({{,textrm{Mod},}}(S))</span> is finitely axiomatizable provided <i>T</i> is finitely axiomatizable. We also prove that <span>({{,textrm{Mod},}}(S))</span> is a quasivariety (variety) provided <i>T</i> is an universal (a positive universal) theory. Some examples are provided.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"107 - 116"},"PeriodicalIF":0.3,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743817","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}

{"title":"Intuitionistic sets and numbers: small set theory and Heyting arithmetic","authors":"Stewart Shapiro,&nbsp;Charles McCarty,&nbsp;Michael Rathjen","doi":"10.1007/s00153-024-00935-4","DOIUrl":"10.1007/s00153-024-00935-4","url":null,"abstract":"<div><p>It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic <span>HA</span>, in a sense to be made precise. Our main target theory, the intuitionistic small set theory <span>SST</span> is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"79 - 105"},"PeriodicalIF":0.3,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00935-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503670","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}

{"title":"The Tarski–Lindenbaum algebra of the class of strongly constructivizable models with (omega )-stable theories","authors":"Mikhail Peretyat’kin","doi":"10.1007/s00153-024-00927-4","DOIUrl":"10.1007/s00153-024-00927-4","url":null,"abstract":"<div><p>We study the class of all strongly constructivizable models having <span>(omega )</span>-stable theories in a fixed finite rich signature. It is proved that the Tarski–Lindenbaum algebra of this class considered together with a Gödel numbering of the sentences is a Boolean <span>(Sigma ^1_1)</span>-algebra whose computable ultrafilters form a dense subset in the set of all ultrafilters; moreover, this algebra is universal with respect to the class of all Boolean <span>(Sigma ^1_1)</span>-algebras. This gives a characterization to the Tarski-Lindenbaum algebra of the class of all strongly constructivizable models with <span>(omega )</span>-stable theories.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"67 - 78"},"PeriodicalIF":0.3,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141368907","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}

{"title":"The Fan Theorem, its strong negation, and the determinacy of games","authors":"Wim Veldman","doi":"10.1007/s00153-024-00930-9","DOIUrl":"10.1007/s00153-024-00930-9","url":null,"abstract":"<div><p>In the context of a weak formal theory called Basic Intuitionistic Mathematics <span>(textsf{BIM})</span>, we study Brouwer’s <i>Fan Theorem</i> and a strong negation of the Fan Theorem, <i>Kleene’s Alternative (to the Fan Theorem)</i>. We prove that the Fan Theorem is equivalent to <i>contrapositions</i> of a number of intuitionistically accepted axioms of countable choice and that Kleene’s Alternative is equivalent to <i>strong negations</i> of these statements. We discuss finite and infinite games and introduce a constructively useful notion of <i>determinacy</i>. We prove that the Fan Theorem is equivalent to the <i>Intuitionistic Determinacy Theorem</i>. This theorem says that every subset of Cantor space <span>(2^omega )</span> is, in our constructively meaningful sense, determinate. Kleene’s Alternative is equivalent to a strong negation of a special case of this theorem. We also consider a <i>uniform intermediate value theorem</i> and a <i>compactness theorem for classical propositional logic</i>. The Fan Theorem is equivalent to each of these theorems and Kleene’s Alternative is equivalent to strong negations of them. We end with a note on <i>‘stronger’</i> Fan Theorems. The paper is a sequel to Veldman (Arch Math Logic 53:621–693, 2014).</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"1 - 66"},"PeriodicalIF":0.3,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00930-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551727","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}

{"title":"Glivenko–Cantelli classes and NIP formulas","authors":"Karim Khanaki","doi":"10.1007/s00153-024-00932-7","DOIUrl":"10.1007/s00153-024-00932-7","url":null,"abstract":"<div><p>We give several new equivalences of <i>NIP</i> for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the <i>NIP</i> context), in an analytic sense. Among other things, we show that for a first order theory <i>T</i> and a formula <span>(phi (x,y))</span>, the following are equivalent: </p><ol>\u0000 <li>\u0000 <span>(i)</span>\u0000 \u0000 <p><span>(phi )</span> has <i>NIP</i> with respect to <i>T</i>.</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>(ii)</span>\u0000 \u0000 <p>For any global <span>(phi )</span>-type <i>p</i>(<i>x</i>) and any model <i>M</i>, if <i>p</i> is finitely satisfiable in <i>M</i>, then <i>p</i> is generalized <i>DBSC</i> definable over <i>M</i>. In particular, if <i>M</i> is countable, then <i>p</i> is <i>DBSC</i> definable over <i>M</i>. (Cf. Definition 3.7, Fact 3.8.)</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>(iii)</span>\u0000 \u0000 <p>For any global Keisler <span>(phi )</span>-measure <span>(mu (x))</span> and any model <i>M</i>, if <span>(mu )</span> is finitely satisfiable in <i>M</i>, then <span>(mu )</span> is generalized Baire-1/2 definable over <i>M</i>. In particular, if <i>M</i> is countable, <span>(mu )</span> is Baire-1/2 definable over <i>M</i>. (Cf. Definition 3.9.)</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>(iv)</span>\u0000 \u0000 <p>For any model <i>M</i> and any Keisler <span>(phi )</span>-measure <span>(mu (x))</span> over <i>M</i>, </p><div><div><span>$$begin{aligned} sup _{bin M}Big |frac{1}{k}sum _{i=1}^kphi (p_i,b)-mu (phi (x,b))Big |rightarrow 0, end{aligned}$$</span></div></div><p> for almost every <span>((p_i)in S_{phi }(M)^{mathbb N})</span> with the product measure <span>(mu ^{mathbb N})</span>. (Cf. Theorem 4.4.)</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>(v)</span>\u0000 \u0000 <p>Suppose moreover that <i>T</i> is countable and <i>NIP</i>, then for any countable model <i>M</i>, the space of global <i>M</i>-finitely satisfied types/measures is a Rosenthal compactum. (Cf. Theorem 5.1.)</p>\u0000 \u0000 </li>\u0000 </ol></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 7-8","pages":"1005 - 1031"},"PeriodicalIF":0.3,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254374","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}

{"title":"Separablilty of metric measure spaces and choice axioms","authors":"Paul Howard","doi":"10.1007/s00153-024-00931-8","DOIUrl":"10.1007/s00153-024-00931-8","url":null,"abstract":"<div><p>In set theory without the Axiom of Choice we prove that the assertion “For every metric space (<i>X</i>, <i>d</i>) with a Borel measure <span>(mu )</span> such that the measure of every open ball is positive and finite, (<i>X</i>, <i>d</i>) is separable.’ is implied by the axiom of choice for countable collections of sets and implies the axiom of choice for countable collections of finite sets. We also show that neither implication is reversible in Zermelo–Fraenkel set theory weakend to permit the existence of atoms and that the second implication is not reversible in Zermelo–Fraenkel set theory. This gives an answer to a question of Dybowski and Górka (Arch Math Logic 62:735–749, 2023. https://doi.org/10.1007/s00153-023-00868-4).</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 7-8","pages":"987 - 1003"},"PeriodicalIF":0.3,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141117609","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}

{"title":"Fragments of IOpen","authors":"Konstantin Kovalyov","doi":"10.1007/s00153-024-00929-2","DOIUrl":"10.1007/s00153-024-00929-2","url":null,"abstract":"<div><p>In this paper we consider some fragments of <span>(textsf{IOpen})</span> (Robinson arithmetic <span>(mathsf Q)</span> with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that <span>(mathsf {I(lit)})</span> is equivalent to <span>(textsf{IOpen})</span> and is not finitely axiomatizable over <span>(mathsf Q)</span>, establish some inclusion relations between <span>(mathsf {I(=)}, mathsf {I(ne )}, mathsf {I(leqslant )})</span> and <span>(textsf{I} (nleqslant ))</span>. We also prove that the set of diophantine equations solvable in models of <span>(mathsf I (=))</span> is (algorithmically) decidable.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 7-8","pages":"969 - 986"},"PeriodicalIF":0.3,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141120436","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}

{"title":"Pathology of submeasures and (F_{sigma }) ideals","authors":"Jorge Martínez,&nbsp;David Meza-Alcántara,&nbsp;Carlos Uzcátegui","doi":"10.1007/s00153-024-00910-z","DOIUrl":"10.1007/s00153-024-00910-z","url":null,"abstract":"<div><p>We address some phenomena about the interaction between lower semicontinuous submeasures on <span>({mathbb {N}})</span> and <span>(F_{sigma })</span> ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological <span>(F_{sigma })</span> ideals. We give a partial answers to the question of whether every nonpathological tall <span>(F_{sigma })</span> ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological <span>(F_{sigma })</span> ideals using sequences in Banach spaces.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 7-8","pages":"941 - 967"},"PeriodicalIF":0.3,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00910-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937160","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}

{"title":"Positive indiscernibles","authors":"Mark Kamsma","doi":"10.1007/s00153-024-00928-3","DOIUrl":"10.1007/s00153-024-00928-3","url":null,"abstract":"<div><p>We generalise various theorems for finding indiscernible trees and arrays to positive logic: based on an existing modelling theorem for s-trees, we prove modelling theorems for str-trees, str<span>(_0)</span>-trees (the reduct of str-trees that forgets the length comparison relation) and arrays. In doing so, we prove stronger versions for basing—rather than locally basing or EM-basing—str-trees on s-trees and str<span>(_0)</span>-trees on str-trees. As an application we show that a thick positive theory has <i>k</i>-<span>(mathsf {TP_2})</span> iff it has 2-<span>(mathsf {TP_2})</span></p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 7-8","pages":"921 - 940"},"PeriodicalIF":0.3,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00928-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937451","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}

{"title":"Spectral MV-algebras and equispectrality","authors":"Giuseppina Gerarda Barbieri,&nbsp;Antonio Di Nola,&nbsp;Giacomo Lenzi","doi":"10.1007/s00153-024-00926-5","DOIUrl":"10.1007/s00153-024-00926-5","url":null,"abstract":"<div><p>In this paper we study the set of MV-algebras with given prime spectrum and we introduce the class of spectral MV-algebras. An MV-algebra is spectral if it is generated by the union of all its prime ideals (or proper ideals, or principal ideals, or maximal ideals). Among spectral MV-algebras, special attention is devoted to bipartite MV-algebras. An MV-algebra is bipartite if it admits an homomorphism onto the MV-algebra of two elements. We prove that both bipartite MV-algebras and spectral MV-algebras can be finitely axiomatized in first order logic. We also prove that there is only, up to isomorphism, a set of MV-algebras with given prime spectrum. A further part of the paper is devoted to some relations between bipartite MV-algebras and their states. Recall that a state on an MV-algebra is a generalization of a probability measure on a Boolean algebra. Particular states are the states with Bayes’ property. We show that an MV-algebra admits a state with the Bayes’ property if and only if it is bipartite.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 7-8","pages":"893 - 919"},"PeriodicalIF":0.3,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00926-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937302","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}