Mathematical Logic Quarterly最新文献

{"title":"The power set and the set of permutations with finitely many non-fixed points of a set","authors":"Guozhen Shen","doi":"10.1002/malq.202100070","DOIUrl":"https://doi.org/10.1002/malq.202100070","url":null,"abstract":"<p>For a cardinal <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>, we write <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mtext>fin</mtext>\u0000 </msub>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$</annotation>\u0000 </semantics></math> for the cardinality of the set of permutations with finitely many non-fixed points of a set which is of cardinality <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>. We investigate the relationships between <math>\u0000 <semantics>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>a</mi>\u0000 </msup>\u0000 <annotation>$2^mathfrak {a}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mtext>fin</mtext>\u0000 </msub>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$</annotation>\u0000 </semantics></math> for an arbitrary infinite cardinal <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math> in <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> (without the axiom of choice). It is proved in <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>a</mi>\u0000 </msup>\u0000 <mo>≠</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mtext>fin</mtext>\u0000 </msub>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$2^mathfrak {a}ne operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$</annotation>\u0000 </semantics></math> for all inf","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"40-45"},"PeriodicalIF":0.3,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50127714","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. 4/2022)","authors":"","doi":"10.1002/malq.202240000","DOIUrl":"10.1002/malq.202240000","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"391-393"},"PeriodicalIF":0.3,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202240000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80624922","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":"Piece selection and cardinal arithmetic","authors":"Pierre Matet","doi":"10.1002/malq.202100033","DOIUrl":"10.1002/malq.202100033","url":null,"abstract":"<p>We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>≥</mo>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>κ</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$lambda ge 2^kappa$</annotation>\u0000 </semantics></math>, then (a) <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$I_{kappa , lambda }$</annotation>\u0000 </semantics></math> is not (λ, 2)-distributive, and (b) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mo>→</mo>\u0000 <msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msubsup>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>ω</mi>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$I_{kappa , lambda }^+ rightarrow (I_{kappa , lambda }^+)^2_omega$</annotation>\u0000 </semantics></math> does not hold.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"416-446"},"PeriodicalIF":0.3,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74802387","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 propositional probability logic","authors":"Anelina Ilić-Stepić,&nbsp;Mateja Knežević,&nbsp;Zoran Ognjanović","doi":"10.1002/malq.202100052","DOIUrl":"10.1002/malq.202100052","url":null,"abstract":"<p>We give a sound and complete axiomatization of a probabilistic extension of intuitionistic logic. Reasoning with probability operators is also intuitionistic (in contradistinction to other works on this topic), i.e., measure functions used for modeling probability operators are partial functions. Finally, we present a decision procedure for our logic, which is a combination of linear programming and an intuitionistic tableaux method.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"479-495"},"PeriodicalIF":0.3,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83342708","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":"Automorphism invariant measures and weakly generic automorphisms","authors":"Gábor Sági","doi":"10.1002/malq.202100044","DOIUrl":"10.1002/malq.202100044","url":null,"abstract":"&lt;p&gt;Let &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be a countable ℵ&lt;sub&gt;0&lt;/sub&gt;-homogeneous structure. The primary motivation of this work is to study different amenability properties of (subgroups of) the automorphism group &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Aut}(mathcal {A})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;; the secondary motivation is to study the existence of weakly generic automorphisms of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Among others, we present sufficient conditions implying the existence of automorphism invariant probability measures on certain subsets of &lt;i&gt;A&lt;/i&gt; and of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Aut}(mathcal {A})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;; we also present sufficient conditions implying that the theory of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is amenable. More concretely, we show that if the set of locally finite automorphisms of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is dense (in particular, if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has weakly generic tuples of automorphisms of arbitrary finite length), then there exists a finitely additive probability measure μ on the subsets of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; definable with parameters such that μ is invariant under &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Aut&lt;/mo&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{Aut}(mathcal {A})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Moreover, if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 4","pages":"458-478"},"PeriodicalIF":0.3,"publicationDate":"2022-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85168891","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":"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":"68 4","pages":"496-515"},"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":"68 4","pages":"447-457"},"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":"5 1","pages":""},"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":"68 4","pages":"398-408"},"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":"68 3","pages":"261-263"},"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}