Mathematical Logic Quarterly最新文献

{"title":"Tameness of definably complete locally o-minimal structures and definable bounded multiplication","authors":"Masato Fujita,&nbsp;Tomohiro Kawakami,&nbsp;Wataru Komine","doi":"10.1002/malq.202200004","DOIUrl":"10.1002/malq.202200004","url":null,"abstract":"<p>We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo-definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo-definable spaces are definably compact. Here, a pseudo-definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set-valued functions with definably compact domains of definition.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89499923","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":"A note on edge colorings and trees","authors":"Adi Jarden,&nbsp;Ziv Shami","doi":"10.1002/malq.202100019","DOIUrl":"10.1002/malq.202100019","url":null,"abstract":"<p>We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors μ satisfies <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>μ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>κ</mi>\u0000 </mrow>\u0000 <annotation>$mu ^+&lt;kappa$</annotation>\u0000 </semantics></math>. Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property, and for each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>,</mo>\u0000 <mi>μ</mi>\u0000 <mo>&lt;</mo>\u0000 <mi>κ</mi>\u0000 </mrow>\u0000 <annotation>$lambda ,mu &lt;kappa$</annotation>\u0000 </semantics></math> there exists <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>κ</mi>\u0000 </mrow>\u0000 <annotation>$kappa ^*&lt;kappa$</annotation>\u0000 </semantics></math> such that every tree of height μ with λ nodes has less than <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <annotation>$kappa ^*$</annotation>\u0000 </semantics></math> branches.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75494728","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":"Controlling the number of normal measures at successor cardinals","authors":"Arthur W. Apter","doi":"10.1002/malq.202000087","DOIUrl":"https://doi.org/10.1002/malq.202000087","url":null,"abstract":"We examine the number of normal measures a successor cardinal can carry, in universes in which the Axiom of Choice is false. When considering successors of singular cardinals, we establish relative consistency results assuming instances of supercompactness, together with the Ultrapower Axiom UA$mathsf {UA}$ (introduced by Goldberg in [12]). When considering successors of regular cardinals, we establish relative consistency results only assuming the existence of one measurable cardinal. This allows for equiconsistencies.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85018665","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":"Extremal numberings and fixed point theorems","authors":"Marat Faizrahmanov","doi":"10.1002/malq.202200035","DOIUrl":"10.1002/malq.202200035","url":null,"abstract":"<p>We consider so-called <i>extremal</i> numberings that form the greatest or minimal degrees under the reducibility of all <i>A</i>-computable numberings of a given family of subsets of <math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$mathbb {N}$</annotation>\u0000 </semantics></math>, where <i>A</i> is an arbitrary oracle. Such numberings are very common in the literature and they are called <i>universal</i> and <i>minimal</i> <i>A</i>-computable numberings, respectively. The main question of this paper is when a universal or a minimal <i>A</i>-computable numbering satisfies the Recursion Theorem (with parameters). First we prove that the Turing degree of a set <i>A</i> is hyperimmune if and only if every universal <i>A</i>-computable numbering satisfies the Recursion Theorem. Next we prove that any universal <i>A</i>-computable numbering satisfies the Recursion Theorem with parameters if <i>A</i> computes a non-computable c.e. set. We also consider the property of precompleteness of universal numberings, which in turn is closely related to the Recursion Theorem. Ershov proved that a numbering is <i>precomplete</i> if and only if it satisfies the Recursion Theorem with parameters for partial computable functions. In this paper, we show that for a given <i>A</i>-computable numbering, in the general case, the Recursion Theorem with parameters for total computable functions is not equivalent to the precompleteness of the numbering, even if it is universal. Finally we prove that if <i>A</i> is high, then any infinite <i>A</i>-computable family has a minimal <i>A</i>-computable numbering that satisfies the Recursion Theorem.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75170334","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":"Contents: (Math. Log. Quart. 3/2022)","authors":"","doi":"10.1002/malq.202230000","DOIUrl":"https://doi.org/10.1002/malq.202230000","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202230000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134806304","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":"Algebraic completion without the axiom of choice","authors":"Jørgen Harmse","doi":"10.1002/malq.202200001","DOIUrl":"10.1002/malq.202200001","url":null,"abstract":"<p>Läuchli and Pincus showed that existence of algebraic completions of all fields cannot be proved from Zermelo-Fraenkel set theory alone. On the other hand, important special cases do follow. In particular, I show that an algebraic completion of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Q</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$mathbb {Q}_p$</annotation>\u0000 </semantics></math> can be constructed in Zermelo-Fraenkel set theory.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78996743","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":"Theory and application of labelling techniques for interpretability logics","authors":"Evan Goris,&nbsp;Marta Bílková,&nbsp;Joost J. Joosten,&nbsp;Luka Mikec","doi":"10.1002/malq.202200015","DOIUrl":"10.1002/malq.202200015","url":null,"abstract":"<p>The notion of a <i>critical successor</i> [5] in relational semantics has been central to most classic modal completeness proofs in interpretability logics. In this paper we shall work with a more general notion, that of an <i>assuring successor</i>. This will enable more concisely formulated completeness proofs, both with respect to ordinary and generalised Veltman semantics. Due to their interesting theoretical properties, we will devote some space to the study of a particular kind of assuring labels, the so-called <i>full labels</i> and <i>maximal labels</i>. After a general treatment of assuringness, we shall apply it to obtain a completeness result for the modal logic <math>\u0000 <semantics>\u0000 <mi>ILP</mi>\u0000 <annotation>$mathsf {ILP}$</annotation>\u0000 </semantics></math> w.r.t. generalised semantics for a restricted class of frames.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81959170","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":"Choiceless large cardinals and set-theoretic potentialism","authors":"Raffaella Cutolo,&nbsp;Joel David Hamkins","doi":"10.1002/malq.202000026","DOIUrl":"10.1002/malq.202000026","url":null,"abstract":"<p>We define a <i>potentialist system</i> of <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math>-structures, i.e., a collection of <i>possible worlds</i> in the language of <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> connected by a binary <i>accessibility relation</i>, achieving a potentialist account of the full background set-theoretic universe <i>V</i>. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math>. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mn>4</mn>\u0000 <mo>.</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$mathsf {S4.2}$</annotation>\u0000 </semantics></math>. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mn>5</mn>\u0000 </mrow>\u0000 <annotation>$mathsf {S5}$</annotation>\u0000 </semantics></math>, both for assertions in the language of <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> and for assertions in the full potentialist language.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77987219","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 the logic of distributive nearlattices","authors":"Luciano J. González","doi":"10.1002/malq.202200012","DOIUrl":"10.1002/malq.202200012","url":null,"abstract":"<p>We study the propositional logic <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>DN</mi>\u0000 </msub>\u0000 <annotation>$mathcal {S}_mathbb {DN}$</annotation>\u0000 </semantics></math> associated with the variety of distributive nearlattices <math>\u0000 <semantics>\u0000 <mi>DN</mi>\u0000 <annotation>$mathbb {DN}$</annotation>\u0000 </semantics></math>. We prove that the logic <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>DN</mi>\u0000 </msub>\u0000 <annotation>$mathcal {S}_mathbb {DN}$</annotation>\u0000 </semantics></math> coincides with the assertional logic associated with the variety <math>\u0000 <semantics>\u0000 <mi>DN</mi>\u0000 <annotation>$mathbb {DN}$</annotation>\u0000 </semantics></math> and with the order-based logic associated with <math>\u0000 <semantics>\u0000 <mi>DN</mi>\u0000 <annotation>$mathbb {DN}$</annotation>\u0000 </semantics></math>. We obtain a characterization of the reduced matrix models of logic <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>DN</mi>\u0000 </msub>\u0000 <annotation>$mathcal {S}_mathbb {DN}$</annotation>\u0000 </semantics></math>. We develop a connection between the logic <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>DN</mi>\u0000 </msub>\u0000 <annotation>$mathcal {S}_mathbb {DN}$</annotation>\u0000 </semantics></math> and the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mo>∧</mo>\u0000 <mo>,</mo>\u0000 <mo>∨</mo>\u0000 <mo>,</mo>\u0000 <mi>⊤</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$lbrace wedge ,vee ,top rbrace$</annotation>\u0000 </semantics></math>-fragment of classical logic. Finally, we present two Hilbert-style axiomatizations for the logic <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>DN</mi>\u0000 </msub>\u0000 <annotation>$mathcal {S}_mathbb {DN}$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77388676","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":"κ-Madness and definability","authors":"Haim Horowitz,&nbsp;Saharon Shelah","doi":"10.1002/malq.202100074","DOIUrl":"10.1002/malq.202100074","url":null,"abstract":"<p>Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal κ, there are no <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>1</mn>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$Sigma ^1_1(kappa )$</annotation>\u0000 </semantics></math> κ-mad families.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76369967","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}