Archive for Mathematical Logic最新文献

{"title":"Different covering numbers of compact tree ideals","authors":"Jelle Mathis Kuiper, Otmar Spinas","doi":"10.1007/s00153-024-00933-6","DOIUrl":"https://doi.org/10.1007/s00153-024-00933-6","url":null,"abstract":"<p>We investigate the covering numbers of some ideals on <span>({^{omega }}{2}{})</span> associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, <span>({mathfrak {C}_{2}})</span>, remain small in the Sacks model.</p>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221700","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":"On categoricity of scattered linear orders of constructive ranks","authors":"Andrey Frolov, Maxim Zubkov","doi":"10.1007/s00153-024-00934-5","DOIUrl":"https://doi.org/10.1007/s00153-024-00934-5","url":null,"abstract":"<p>In this article we investigate the complexity of isomorphisms between scattered linear orders of constructive ranks. We give the general upper bound and prove that this bound is sharp. Also, we construct examples showing that the categoricity level of a given scattered linear order can be an arbitrary ordinal from 3 to the upper bound, except for the case when the ordinal is the successor of a limit ordinal. The existence question of the scattered linear orders whose categoricity level equals the successor of a limit ordinal is still open.\u0000</p>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221698","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 provably total functions of basic arithmetic and its extensions","authors":"Mohammad Ardeshir, Erfan Khaniki, Mohsen Shahriari","doi":"10.1007/s00153-024-00939-0","DOIUrl":"https://doi.org/10.1007/s00153-024-00939-0","url":null,"abstract":"<p>We study Basic Arithmetic, <span>(textsf{BA})</span> introduced by Ruitenburg (Notre Dame J Formal Logic 39:18–46, 1998). <span>(textsf{BA})</span> is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of <span>(textsf{BA})</span> is a <i>proper</i> sub-class of the primitive recursive functions. Three extensions of <span>(textsf{BA})</span>, called <span>(textsf{BA}+mathsf U)</span>, <span>(mathsf {BA_{mathrm c}})</span> and <span>(textsf{EBA})</span> are investigated with relation to their provably total recursive functions. It is shown that the provably total recursive functions of these three extensions of <span>(textsf{BA})</span> are <i>exactly</i> the primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem does not hold in <span>(textsf{BA})</span>, <span>(textsf{BA}+mathsf U)</span>, <span>(mathsf {BA_{mathrm c}})</span>, but holds in <span>(textsf{EBA})</span>.</p>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221699","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":"Undecidability of indecomposable polynomial rings","authors":"Marco Barone, Nicolás Caro-Montoya, Eudes Naziazeno","doi":"10.1007/s00153-024-00936-3","DOIUrl":"https://doi.org/10.1007/s00153-024-00936-3","url":null,"abstract":"<p>By using algebraic properties of (commutative unital) indecomposable polynomial rings we achieve results concerning their first-order theory, namely: interpretability of arithmetic and a uniform proof of undecidability of their full theory, both in the language of rings without parameters. This vastly extends the scope of a method due to <span>Raphael Robinson</span>, which deals with a restricted class of polynomial integral domains.</p>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141940768","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":"On some $$Sigma ^{B}_{0}$$ -formulae generalizing counting principles over $$V^{0}$$","authors":"Eitetsu Ken","doi":"10.1007/s00153-024-00938-1","DOIUrl":"https://doi.org/10.1007/s00153-024-00938-1","url":null,"abstract":"<p>We formalize various counting principles and compare their strengths over <span>(V^{0})</span>. In particular, we conjecture the following mutual independence between:</p><ul>\u0000<li>\u0000<p>a uniform version of modular counting principles and the pigeonhole principle for injections,</p>\u0000</li>\u0000<li>\u0000<p>a version of the oddtown theorem and modular counting principles of modulus <i>p</i>, where <i>p</i> is any natural number which is not a power of 2,</p>\u0000</li>\u0000<li>\u0000<p>and a version of Fisher’s inequality and modular counting principles.</p>\u0000</li>\u0000</ul><p> Then, we give sufficient conditions to prove them. We give a variation of the notion of <i>PHP</i>-tree and <i>k</i>-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have <i>o</i>(<i>n</i>)-evaluations. As for the remaining two, we utilize well-known notions of <i>p</i>-tree and <i>k</i>-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.</p>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784682","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":"On absorption’s formula definable semigroups of complete theories","authors":"Mahsut Bekenov, Aida Kassatova, Anvar Nurakunov","doi":"10.1007/s00153-024-00937-2","DOIUrl":"https://doi.org/10.1007/s00153-024-00937-2","url":null,"abstract":"<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>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"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, Charles McCarty, Michael Rathjen","doi":"10.1007/s00153-024-00935-4","DOIUrl":"https://doi.org/10.1007/s00153-024-00935-4","url":null,"abstract":"<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>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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":"https://doi.org/10.1007/s00153-024-00930-9","url":null,"abstract":"<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>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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":"https://doi.org/10.1007/s00153-024-00932-7","url":null,"abstract":"<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<p><span>(phi )</span> has <i>NIP</i> with respect to <i>T</i>.</p>\u0000</li>\u0000<li>\u0000<span>(ii)</span>\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</li>\u0000<li>\u0000<span>(iii)</span>\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</li>\u0000<li>\u0000<span>(iv)</span>\u0000<p>For any model <i>M</i> and any Keisler <span>(phi )</span>-measure <span>(mu (x))</span> over <i>M</i>, </p><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><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</li>\u0000<li>\u0000<span>(v)</span>\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</li>\u0000</ol>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"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":"Pathology of submeasures and $$F_{sigma }$$ ideals","authors":"Jorge Martínez, David Meza-Alcántara, Carlos Uzcátegui","doi":"10.1007/s00153-024-00910-z","DOIUrl":"https://doi.org/10.1007/s00153-024-00910-z","url":null,"abstract":"<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>","PeriodicalId":8350,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}