{"title":"THEOREMS OF HYPERARITHMETIC ANALYSIS AND ALMOST THEOREMS OF HYPERARITHMETIC ANALYSIS","authors":"James S. Barnes, Jun Le Goh, R. Shore","doi":"10.1017/bsl.2021.70","DOIUrl":"https://doi.org/10.1017/bsl.2021.70","url":null,"abstract":"Abstract Theorems of hyperarithmetic analysis (THAs) occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed (recursive) iterations of the Turing jump but below ATR \u0000$_{0}$\u0000 (and so \u0000$Pi _{1}^{1}$\u0000 -CA \u0000$_{0}$\u0000 or the hyperjump). There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They seem to be typical applications of ACA \u0000$_{0}$\u0000 but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity. This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis (ATHAs). When combined with ACA \u0000$_{0}$\u0000 they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes ( \u0000$Pi _{1}^{1}$\u0000 , r- \u0000$Pi _{1}^{1}$\u0000 , and Tanaka) are defined and these ATHAs as well as many other principles are shown to be conservative over RCA \u0000$_{0}$\u0000 in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA \u0000$_{0}$\u0000 but over ACA \u0000$_{0}$\u0000 is equivalent to the other which may be strong (THA) or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"167 1","pages":"133 - 149"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90429071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"SATURATED MODELS FOR THE WORKING MODEL THEORIST","authors":"Yatir Halevi, Itay Kaplan","doi":"10.1017/bsl.2023.6","DOIUrl":"https://doi.org/10.1017/bsl.2023.6","url":null,"abstract":"Abstract We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"63 1","pages":"163 - 169"},"PeriodicalIF":0.0,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89176666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"INVARIANCE CRITERIA AS META-CONSTRAINTS","authors":"Gil Sagi","doi":"10.1017/bsl.2021.67","DOIUrl":"https://doi.org/10.1017/bsl.2021.67","url":null,"abstract":"Abstract Invariance criteria are widely accepted as a means to demarcate the logical vocabulary of a language. In previous work, I proposed a framework of “semantic constraints” for model-theoretic consequence which does not rely on a strict distinction between logical and nonlogical terms, but rather on a range of constraints on models restricting the interpretations of terms in the language in different ways. In this paper I show how invariance criteria can be generalized so as to apply to semantic constraints on models. Some obviously unpalatable semantic constraints turn out to be invariant under isomorphisms. I shall connect the discussion to known counter-examples to invariance criteria for logical terms, and so the generalization will also shed light on the current existing debate on logicality. I analyse the failure of invariance to fulfil its role as a criterion for logicality, and argue that invariance conditions should best be thought of as merely methodological meta-constraints restricting the ways the model-theoretic apparatus should be used.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"115 1","pages":"104 - 132"},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79554184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Giulio Fellin, P. Schuster, Daniel Misselbeck-Wessel
{"title":"THE JACOBSON RADICAL OF A PROPOSITIONAL THEORY","authors":"Giulio Fellin, P. Schuster, Daniel Misselbeck-Wessel","doi":"10.1017/bsl.2021.66","DOIUrl":"https://doi.org/10.1017/bsl.2021.66","url":null,"abstract":"Abstract Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment relation over a single-conclusion one.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"157 1","pages":"163 - 181"},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87917425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"New Directions in Duality Theory for Modal Logic","authors":"L. Carai","doi":"10.1017/bsl.2021.52","DOIUrl":"https://doi.org/10.1017/bsl.2021.52","url":null,"abstract":"Abstract In this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers. Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean \u0000$ell $\u0000 -algebras to a duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean \u0000$ell $\u0000 -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic. Abstract taken directly from the thesis. E-mail: lcarai@unisa.it URL: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&cbl=18750&diss=y","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"22 1","pages":"527 - 527"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84405099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Second-Order Modal Logic","authors":"Andrew Parisi","doi":"10.1017/bsl.2020.45","DOIUrl":"https://doi.org/10.1017/bsl.2020.45","url":null,"abstract":"Abstract The dissertation introduces new sequent-calculi for free first- and second-order logic, and a hyper-sequent calculus for modal logics K, D, T, B, S4, and S5; to attain the calculi for the stronger modal logics, only external structural rules need to be added to the calculus for K, while operational and internal structural rules remain the same. Completeness and cut-elimination are proved for all calculi presented. Philosophically, the dissertation develops an inferentialist, or proof-theoretic, theory of meaning. It takes as a starting point that the sense of a sentence is determined by the rules governing its use. In particular, there are two features of the use of a sentence that jointly determine its sense, the conditions under which it is coherent to assert that sentence and the conditions under which it is coherent to deny that sentence. The dissertation develops a theory of quantification as marking coherent ways a language can be expanded and modality as the means by which we can reflect on the norms governing the assertion and denial conditions of our language. If the view of quantification that is argued for is correct, then there is no tension between second-order quantification and nominalism. In particular, the ontological commitments one can incur through the use of a quantifier depend wholly on the ontological commitments one can incur through the use of atomic sentences. The dissertation concludes by applying the developed theory of meaning to the metaphysical issue of necessitism and contingentism. Two objections to a logic of contingentism are raised and addressed. The resulting logic is shown to meet all the requirement that the dissertation lays out for a theory of meaning for quantifiers and modal operators. Abstract prepared by Andrew Parisi E-mail: andrew.p.parisi@gmail.com URL: https://opencommons.uconn.edu/dissertations/1480/","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"25 1","pages":"530 - 531"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88499404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dimensions of Ordinals: Set Theory, Homology Theory, and the First Omega Alephs","authors":"J. Bergfalk","doi":"10.1017/bsl.2021.36","DOIUrl":"https://doi.org/10.1017/bsl.2021.36","url":null,"abstract":"Abstract We describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. It describes ZFC principles distinguishing among the ordinals of the form \u0000$omega _n$\u0000 . More precisely, this framework correlates each \u0000$omega _n$\u0000 with an \u0000$(n+1)$\u0000 -dimensional generalization of Todorcevic’s walks technique, and begins to account for that technique’s “unreasonable effectiveness” on \u0000$omega _1$\u0000 . We show in contrast that on higher cardinals \u0000$kappa $\u0000 , the existence of these principles is frequently independent of the ZFC axioms. Finally, we detail implications of these phenomena for the computation of strong homology groups and higher derived limits, deriving independence results in algebraic topology and homological algebra, respectively, in the process. Abstract prepared by Jeffrey Bergfalk. E-mail: jeffrey.bergfalk@univie.ac.at","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"21 1","pages":"526 - 527"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75891723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"VARIANTS OF KREISEL’S CONJECTURE ON A NEW NOTION OF PROVABILITY","authors":"P. G. Santos, R. Kahle","doi":"10.1017/bsl.2021.68","DOIUrl":"https://doi.org/10.1017/bsl.2021.68","url":null,"abstract":"Abstract Kreisel’s conjecture is the statement: if, for all \u0000$nin mathbb {N}$\u0000 , \u0000$mathop {text {PA}} nolimits vdash _{k text { steps}} varphi (overline {n})$\u0000 , then \u0000$mathop {text {PA}} nolimits vdash forall x.varphi (x)$\u0000 . For a theory of arithmetic T, given a recursive function h, \u0000$T vdash _{leq h} varphi $\u0000 holds if there is a proof of \u0000$varphi $\u0000 in T whose code is at most \u0000$h(#varphi )$\u0000 . This notion depends on the underlying coding. \u0000${P}^h_T(x)$\u0000 is a predicate for \u0000$vdash _{leq h}$\u0000 in T. It is shown that there exist a sentence \u0000$varphi $\u0000 and a total recursive function h such that \u0000$Tvdash _{leq h}mathop {text {Pr}} nolimits _T(ulcorner mathop {text {Pr}} nolimits _T(ulcorner varphi urcorner )rightarrow varphi urcorner )$\u0000 , but , where \u0000$mathop {text {Pr}} nolimits _T$\u0000 stands for the standard provability predicate in T. This statement is related to a conjecture by Montagna. Also variants and weakenings of Kreisel’s conjecture are studied. By the use of reflexion principles, one can obtain a theory \u0000$T^h_Gamma $\u0000 that extends T such that a version of Kreisel’s conjecture holds: given a recursive function h and \u0000$varphi (x)$\u0000 a \u0000$Gamma $\u0000 -formula (where \u0000$Gamma $\u0000 is an arbitrarily fixed class of formulas) such that, for all \u0000$nin mathbb {N}$\u0000 , \u0000$Tvdash _{leq h} varphi (overline {n})$\u0000 , then \u0000$T^h_Gamma vdash forall x.varphi (x)$\u0000 . Derivability conditions are studied for a theory to satisfy the following implication: if , then \u0000$Tvdash forall x.varphi (x)$\u0000 . This corresponds to an arithmetization of Kreisel’s conjecture. It is shown that, for certain theories, there exists a function h such that \u0000$vdash _{k text { steps}} subseteq vdash _{leq h}$\u0000 .","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"29 1","pages":"337 - 350"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91154353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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