{"title":"A dichotomy for \u0000 \u0000 T\u0000 $T$\u0000 -convex fields with a monomial group","authors":"Elliot Kaplan, Christoph Kesting","doi":"10.1002/malq.202300017","DOIUrl":"10.1002/malq.202300017","url":null,"abstract":"<p>We prove a dichotomy for o-minimal fields <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathcal {R}$</annotation>\u0000 </semantics></math>, expanded by a <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math>-convex valuation ring (where <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math> is the theory of <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathcal {R}$</annotation>\u0000 </semantics></math>) and a compatible monomial group. We show that if <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math> is power bounded, then this expansion of <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathcal {R}$</annotation>\u0000 </semantics></math> is model complete (assuming that <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$T$</annotation>\u0000 </semantics></math> is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathcal {R}$</annotation>\u0000 </semantics></math> defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model-theoretic tameness.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 1","pages":"99-110"},"PeriodicalIF":0.3,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526085","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 persistence principle over weak interpretability logic","authors":"Sohei Iwata, Taishi Kurahashi, Yuya Okawa","doi":"10.1002/malq.202200020","DOIUrl":"10.1002/malq.202200020","url":null,"abstract":"<p>We focus on the persistence principle over weak interpretability logic. Our object of study is the logic obtained by adding the persistence principle to weak interpretability logic from several perspectives. Firstly, we prove that this logic enjoys a weak version of the fixed point property. Secondly, we introduce a system of sequent calculus and prove the cut-elimination theorem for it. As a consequence, we prove that the logic enjoys the Craig interpolation property. Thirdly, we show that the logic is the natural basis of a generalization of simplified Veltman semantics, and prove that it has the finite frame property with respect to that semantics. Finally, we prove that it is sound and complete with respect to some appropriate arithmetical semantics.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 1","pages":"37-63"},"PeriodicalIF":0.3,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136317155","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":"Infinite Wordle and the mastermind numbers","authors":"Joel David Hamkins","doi":"10.1002/malq.202200049","DOIUrl":"10.1002/malq.202200049","url":null,"abstract":"<p>I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game-theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of <i>n</i> letters, including infinite words or even uncountable words, the codebreaker can nevertheless always win in <i>n</i> steps. Meanwhile, the <i>mastermind number</i> <math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$mathbbm {m}$</annotation>\u0000 </semantics></math>, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length ω over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of <math>\u0000 <semantics>\u0000 <mi>ZFC</mi>\u0000 <annotation>$mathsf {ZFC}$</annotation>\u0000 </semantics></math>, for it is provably equal to the eventually different number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mo>≠</mo>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathfrak {d}({ne ^*})$</annotation>\u0000 </semantics></math>, which is the same as the covering number of the meager ideal <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mtext>cov</mtext>\u0000 </mrow>\u0000 <mo>(</mo>\u0000 <mi>M</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{mbox{cov}}(mathcal {M})$</annotation>\u0000 </semantics></math>. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 4","pages":"400-416"},"PeriodicalIF":0.3,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783992","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":"When cardinals determine the power set: inner models and Härtig quantifier logic","authors":"Jouko Väänänen, Philip D. Welch","doi":"10.1002/malq.202200030","DOIUrl":"10.1002/malq.202200030","url":null,"abstract":"<p>We show that the predicate “<i>x</i> <i>is the power set of</i> <i>y</i>” is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mo>Card</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$Sigma _1(operatorname{Card})$</annotation>\u0000 </semantics></math>-definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here <math>\u0000 <semantics>\u0000 <mo>Card</mo>\u0000 <annotation>$operatorname{Card}$</annotation>\u0000 </semantics></math> is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>V</mi>\u0000 <mi>I</mi>\u0000 </msub>\u0000 <annotation>$V_I$</annotation>\u0000 </semantics></math>, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model has a cardinal strong up to one of its ℵ-fixed points, and <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>I</mi>\u0000 </msub>\u0000 <annotation>$ell _{I}$</annotation>\u0000 </semantics></math>, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>I</mi>\u0000 </msub>\u0000 <annotation>$ell _I$</annotation>\u0000 </semantics></math> is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 4","pages":"460-471"},"PeriodicalIF":0.3,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136024402","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":"A classification of low c.e. sets and the Ershov hierarchy","authors":"Marat Faizrahmanov","doi":"10.1002/malq.202300020","DOIUrl":"10.1002/malq.202300020","url":null,"abstract":"<p>In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ-levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ-level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Sigma ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>- and <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Delta ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>-bound for every infinite computable ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Sigma ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>- or <math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Delta ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>-bound of a c.e. set <i>A</i> is equivalent to the fact that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>A</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <mo>∈</mo>\u0000 <msubsup>\u0000 <mi>Δ</mi>\u0000 <mi>α</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$A^{prime }in Delta ^{-1}_{alpha }$</annotation>\u0000 </semantics></math>. Finally, we consider the generalized truth-table reducibilities <math>\u0000 <semantics>\u0000 <msub>\u0000 <mo>⩽</mo>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mi>t</mi>\u0000 <mi>t</mi>\u0000 <mo>(</mo>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 4","pages":"508-518"},"PeriodicalIF":0.3,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981501","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":"Approximate isomorphism of metric structures","authors":"James E. Hanson","doi":"10.1002/malq.202200076","DOIUrl":"10.1002/malq.202200076","url":null,"abstract":"<p>We give a formalism for approximate isomorphism in continuous logic simultaneously generalizing those of two papers by Ben Yaacov [2] and by Ben Yaacov, Doucha, Nies, and Tsankov [6], which are largely incompatible. With this we explicitly exhibit Scott sentences for the perturbation systems of the former paper, such as the Banach-Mazur distance and the Lipschitz distance between metric spaces. Our formalism is simultaneously characterized syntactically by a mild generalization of perturbation systems and semantically by certain elementary classes of two-sorted structures that witness approximate isomorphism. As an application, we show that the theory of any <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-tree or ultrametric space of finite radius is stable, improving a result of Carlisle and Henson [8].</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 4","pages":"482-507"},"PeriodicalIF":0.3,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200076","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84346103","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":"Pregeometry over locally o-minimal structures and dimension","authors":"Masato Fujita","doi":"10.1002/malq.202200069","DOIUrl":"10.1002/malq.202200069","url":null,"abstract":"<p>We define a discrete closure operator for definably complete locally o-minimal structures <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math>. The pair of the underlying set of <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it <math>\u0000 <semantics>\u0000 <mo>discl</mo>\u0000 <annotation>$operatorname{discl}$</annotation>\u0000 </semantics></math>-dimension. A definable set <i>X</i> is of dimension equal to the <math>\u0000 <semantics>\u0000 <mo>discl</mo>\u0000 <annotation>$operatorname{discl}$</annotation>\u0000 </semantics></math>-dimension of <i>X</i>. The structure <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> also coincides with its dimension.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 4","pages":"472-481"},"PeriodicalIF":0.3,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76113023","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}
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