G. Bezhanishvili, F. Dashiell Jr, A. Moshier, J. Walters-Wayland
{"title":"Degrees of join-distributivity via Bruns-Lakser towers","authors":"G. Bezhanishvili, F. Dashiell Jr, A. Moshier, J. Walters-Wayland","doi":"arxiv-2409.04894","DOIUrl":"https://doi.org/arxiv-2409.04894","url":null,"abstract":"We utilize the Bruns-Lakser completion to introduce Bruns-Lakser towers of a\u0000meet-semilattice. This machinery enables us to develop various hierarchies\u0000inside the class of bounded distributive lattices, which measure\u0000$kappa$-degrees of distributivity of bounded distributive lattices and their\u0000Dedekind-MacNeille completions. We also use Priestley duality to obtain a dual\u0000characterization of the resulting hierarchies. Among other things, this yields\u0000a natural generalization of Esakia's representation of Heyting lattices to\u0000proHeyting lattices.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187732","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":"Transposition of variables is hard to describe","authors":"H. Andréka, I. Németi, Zs. Tuza","doi":"arxiv-2409.04088","DOIUrl":"https://doi.org/arxiv-2409.04088","url":null,"abstract":"The function $p_{xy}$ that interchanges two logical variables $x,y$ in\u0000formulas is hard to describe in the following sense. Let $F$ denote the\u0000Lindenbaum-Tarski formula-algebra of a finite-variable first order logic,\u0000endowed with $p_{xy}$ as a unary function. Each equational axiom system for the\u0000equational theory of $F$ has to contain, for each finite $n$, an equation that\u0000contains together with $p_{xy}$ at least $n$ algebraic variables, and each of\u0000the operations $exists, =, lor$. This solves a problem raised by Johnson [J.\u0000Symb. Logic] more than 50 years ago: the class of representable polyadic\u0000equality algebras of a finite dimension $nge 3$ cannot be axiomatized by\u0000adding finitely many equations to the equational theory of representable\u0000cylindric algebras of dimension $n$. Consequences for proof systems of\u0000finite-variable logic and for defining equations of polyadic equality algebras\u0000are given.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187738","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}
Itaï Ben Yaacov, Pablo Destic, Ehud Hrushovski, Michał Szachniewicz
{"title":"Globally valued fields: foundations","authors":"Itaï Ben Yaacov, Pablo Destic, Ehud Hrushovski, Michał Szachniewicz","doi":"arxiv-2409.04570","DOIUrl":"https://doi.org/arxiv-2409.04570","url":null,"abstract":"We present foundations of globally valued fields, i.e., of a class of fields\u0000with an extra structure, capturing some aspects of the geometry of global\u0000fields, based on the product formula. We provide a dictionary between various\u0000data defining such extra structure: syntactic (models of some unbounded\u0000continuous logic theory), Arakelov theoretic, and measure theoretic. In\u0000particular we obtain a representation theorem relating globally valued fields\u0000and adelic curves defined by Chen and Moriwaki.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187733","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":"On Classical Determinate Truth","authors":"Luca Castaldo, Carlo Nicolai","doi":"arxiv-2409.04316","DOIUrl":"https://doi.org/arxiv-2409.04316","url":null,"abstract":"{The paper studies classical, type-free theories of truth and\u0000determinateness. Recently, Volker Halbach and Kentaro Fujimoto proposed a novel\u0000approach to classical determinate truth, in which determinateness is\u0000axiomatized by a primitive predicate. In the paper we propose a different\u0000strategy to develop theories of classical determinate truth in Halbach and\u0000Fujimoto's sense featuring a emph{defined} determinateness predicate. This\u0000puts our theories of classical determinate truth in continuity with a standard\u0000approach to determinateness by authors such as Feferman and Reinhardt. The\u0000theories entail all principles of Fujimoto and Halbach's theories, and are\u0000proof-theoretically equivalent to Halbach and Fujimoto's CD+. They will be\u0000shown to be logically equivalent to a class of natural theories of truth, the\u0000emph{classical closures of Kripke-Feferman truth}. The analysis of the\u0000proposed theories will also provide new insights on Fujimoto and Halbach's\u0000theories: we show that the latter cannot prove most of the axioms of the\u0000classical closures of Kripke-Feferman truth. This entails that, unlike what\u0000happens in our theories of truth and determinateness, Fujimoto and Halbach's\u0000emph{inner theories} -- the sentences living under two layers of truth --\u0000cannot be closed under standard logical rules of inference.}","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187735","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":"Reflections on Russell's antinomy","authors":"Paola Cattabriga","doi":"arxiv-2409.05903","DOIUrl":"https://doi.org/arxiv-2409.05903","url":null,"abstract":"We present Russell's antinomy using three distinct deductive systems, which\u0000are then compared to deepen the logical deductions that lead to the\u0000contradiction. Some inferential paths are then presented, alternative to the\u0000commonly accepted one, that allow for the formal assertion of the antinomy\u0000without deriving the contradiction, thus preserving the coherence of the\u0000system. In light of this, the purpose of this article is to propose a review of\u0000the consequences of asserting Russell's antinomy and, by extension, the\u0000widespread belief that any attempt to resolve a paradox is doomed to failure.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187709","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":"Coding is hard","authors":"Sam Sanders","doi":"arxiv-2409.04562","DOIUrl":"https://doi.org/arxiv-2409.04562","url":null,"abstract":"A central topic in mathematical logic is the classification of theorems from\u0000mathematics in hierarchies according to their logical strength. Ideally, the\u0000place of a theorem in a hierarchy does not depend on the representation (aka\u0000coding) used. In this paper, we show that the standard representation of\u0000compact metric spaces in second-order arithmetic has a profound effect. To this\u0000end, we study basic theorems for such spaces like a continuous function has a\u0000supremum and a countable set has measure zero. We show that these and similar\u0000third-order statements imply at least Feferman's highly non-constructive\u0000projection principle, and even full second-order arithmetic or countable choice\u0000in some cases. When formulated with representations (aka codes), the associated\u0000second-order theorems are provable in rather weak fragments of second-order\u0000arithmetic. Thus, we arrive at the slogan that coding compact metric spaces in\u0000the language of second-order arithmetic can be as hard as second-order\u0000arithmetic or countable choice. We believe every mathematician should be aware\u0000of this slogan, as central foundational topics in mathematics make use of the\u0000standard second-order representation of compact metric spaces. In the process\u0000of collecting evidence for the above slogan, we establish a number of\u0000equivalences involving Feferman's projection principle and countable choice. We\u0000also study generalisations to fourth-order arithmetic and beyond with\u0000similar-but-stronger results.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187734","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":"The Kaufmann--Clote question on end extensions of models of arithmetic and the weak regularity principle","authors":"Mengzhou Sun","doi":"arxiv-2409.03527","DOIUrl":"https://doi.org/arxiv-2409.03527","url":null,"abstract":"We investigate the end extendibility of models of arithmetic with restricted\u0000elementarity. By utilizing the restricted ultrapower construction in the\u0000second-order context, for each $ninmathbb{N}$ and any countable model of\u0000$mathrm{B}Sigma_{n+2}$, we construct a proper $Sigma_{n+2}$-elementary end\u0000extension satisfying $mathrm{B}Sigma_{n+1}$, which answers a question by\u0000Clote positively. We also give a characterization of countable models of\u0000$mathrm{I}Sigma_{n+2}$ in terms of their end extendibility similar to the\u0000case of $mathrm{B}Sigma_{n+2}$. Along the proof, we will introduce a new type\u0000of regularity principles in arithmetic called the weak regularity principle,\u0000which serves as a bridge between the model's end extendibility and the amount\u0000of induction or collection it satisfies.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187737","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":"Model theory on Hilbert spaces expanded by a representation of a group","authors":"Alexander Berenstein, Juan Manuel Pérez","doi":"arxiv-2409.03923","DOIUrl":"https://doi.org/arxiv-2409.03923","url":null,"abstract":"In this paper we study expansions of infinite dimensional Hilbert spaces with\u0000a unitary representation of a group. When the group is finite, we prove the\u0000theory of the corresponding expansion is $aleph_0$-categorical,\u0000$aleph_0$-stable and is SFB. On the other hand, when the group involved is a\u0000product of the form $Htimes mathbb{Z}^n$, where $H$ is a finite group and\u0000$ngeq 1$, the theory of the Hilbert space expanded by the representation of\u0000this group is, in general, stable not $aleph_0$-stable, not\u0000$aleph_0$-categorical, but it is $aleph_0$-categorical up to perturbations\u0000and $aleph_0$-stable up to perturbations.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227676","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":"Forcing as a Local Method of Accessing Small Extensions","authors":"Desmond Lau","doi":"arxiv-2409.03441","DOIUrl":"https://doi.org/arxiv-2409.03441","url":null,"abstract":"Fix a set-theoretic universe $V$. We look at small extensions of $V$ as\u0000generalised degrees of computability over $V$. We also formalise and\u0000investigate the complexity of certain methods one can use to define, in $V$,\u0000subclasses of degrees over $V$. Finally, we give a nice characterisation of the\u0000complexity of forcing within this framework.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224554","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":"On the universal theory of the free pseudocomplemented distributive lattice","authors":"Luca Carai, Tommaso Moraschini","doi":"arxiv-2409.03640","DOIUrl":"https://doi.org/arxiv-2409.03640","url":null,"abstract":"It is shown that the universal theory of the free pseudocomplemented\u0000distributive lattice is decidable and a recursive axiomatization is presented.\u0000This contrasts with the case of the full elementary theory of the finitely\u0000generated free algebras which is known to be undecidable. As a by-product, a\u0000description of the pseudocomplemented distributive lattices that can be\u0000embedded into the free algebra is also obtained.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187736","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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