{"title":"Formal model theory and higher topology","authors":"Ivan Di Liberti","doi":"10.1002/malq.202300006","DOIUrl":"10.1002/malq.202300006","url":null,"abstract":"<p>We study the 2-categories BIon, of (generalized) bounded ionads, and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mtext>Acc</mtext>\u0000 <mi>ω</mi>\u0000 </msub>\u0000 <annotation>$text{Acc}_omega$</annotation>\u0000 </semantics></math>, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex) geometric sketches, which serve as categorical specifications of geometric theories. We provide reconstruction and completeness-like results. We relate abstract elementary classes to locally decidable topoi. We introduce the notion of categories of saturated objects and relate it to atomic topoi.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141167709","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 property of forcing notions and preservation of cardinal invariants","authors":"Yushiro Aoki","doi":"10.1002/malq.202300013","DOIUrl":"10.1002/malq.202300013","url":null,"abstract":"<p>We define a property of forcing notions and show that there exists a model of its forcing axiom and the negation of the continuum hypothesis in which the Cichoń-Blass diagram of cardinal invariants is the same as in the Cohen model. As a corollary, its forcing axiom and the forcing axiom for <span></span><math>\u0000 <semantics>\u0000 <mi>σ</mi>\u0000 <annotation>$sigma$</annotation>\u0000 </semantics></math>-centered forcing notions are independent of each other.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526111","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":"Infinitary logic with infinite sequents: syntactic investigations","authors":"Matteo Tesi","doi":"10.1002/malq.202300011","DOIUrl":"10.1002/malq.202300011","url":null,"abstract":"<p>The present paper deals with a purely syntactic analysis of infinitary logic with infinite sequents. In particular, we discuss sequent calculi for classical and intuitionistic infinitary logic with good structural properties based on sequents possibly containing infinitely many formulas. A cut admissibility proof is proposed which employs a new strategy and a new inductive parameter. We conclude the paper by discussing related issues and possible themes for future research.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138526095","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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