Pub Date : 2021-05-01DOI: 10.1215/00294527-2021-0012
Andreas Fjellstad, Jan-Fredrik Olsen
In this note, we show that the first-order logic IKω is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKTω obtained by expanding IKω with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for IKTω depends on how vacuous quantification is treated.
{"title":"IKTω and Łukasiewicz-Models","authors":"Andreas Fjellstad, Jan-Fredrik Olsen","doi":"10.1215/00294527-2021-0012","DOIUrl":"https://doi.org/10.1215/00294527-2021-0012","url":null,"abstract":"In this note, we show that the first-order logic IKω is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKTω obtained by expanding IKω with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for IKTω depends on how vacuous quantification is treated.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87275547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-01DOI: 10.1215/00294527-2021-0017
Chris J. Conidis
We construct a computable module M over a computable commutative ring R such that the radical of M, rad(M), defined as the intersection of all proper maximal submodules, is Π1-complete. This shows that in general such radicals are as (logically) complicated as possible and, unlike many other kinds of ring-theoretic radicals, admit no arithmetical definition.
{"title":"The Complexity of Module Radicals","authors":"Chris J. Conidis","doi":"10.1215/00294527-2021-0017","DOIUrl":"https://doi.org/10.1215/00294527-2021-0017","url":null,"abstract":"We construct a computable module M over a computable commutative ring R such that the radical of M, rad(M), defined as the intersection of all proper maximal submodules, is Π1-complete. This shows that in general such radicals are as (logically) complicated as possible and, unlike many other kinds of ring-theoretic radicals, admit no arithmetical definition.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78015699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-01DOI: 10.1215/00294527-2021-0020
Sergi Oms, E. Zardini
{"title":"Inclosure and Intolerance","authors":"Sergi Oms, E. Zardini","doi":"10.1215/00294527-2021-0020","DOIUrl":"https://doi.org/10.1215/00294527-2021-0020","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86699136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-23DOI: 10.1215/00294527-2022-0022
Christopher J. Eagle, Alan Getz
We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered in Fräıssé theory.
{"title":"Model-Theoretic Properties of Dynamics on the Cantor Set","authors":"Christopher J. Eagle, Alan Getz","doi":"10.1215/00294527-2022-0022","DOIUrl":"https://doi.org/10.1215/00294527-2022-0022","url":null,"abstract":"We examine topological dynamical systems on the Cantor set from the point of view of the continuous model theory of commutative C*-algebras. After some general remarks we focus our attention on the generic homeomorphism of the Cantor set, as constructed by Akin, Glasner, and Weiss. We show that this homeomorphism is the prime model of its theory. We also show that the notion of “generic” used by Akin, Glasner, and Weiss is distinct from the notion of “generic” encountered in Fräıssé theory.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43916848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-14DOI: 10.1215/00294527-2021-0034
J. Moschovakis
In 2002 Robert Solovay proved that a subsystem BI of classical second order arithmetic, with bar induction and arithmetical countable choice, can be negatively interpreted in the neutral subsystem BSK of Kleene's intuitionistic analysis FIM using Markov's Principle MP. Combining this result with Kleene's formalized recursive realizability, he established (in primitive recursive arithmetic PRA) that FIM + MP and BI have the same consistency strength. This historical note includes Solovay's original proof, with his permission, and the additional observation that Markov's Principle can be weakened to a double negation shift axiom consistent with Brouwer's creating subject counterexamples.
Robert Solovay(2002)利用Markov原理证明了具有条形归纳和算术可数选择的经典二阶算法的子系统BI可以用Kleene的直觉分析FIM的中性子系统BSK负解释。结合Kleene的形式化递归可实现性，他建立了(在原始递归算法PRA中)FIM + MP与BI具有相同的一致性强度。这篇历史笔记包括Solovay的原始证明，在他的允许下，以及马尔可夫原理可以被削弱为双重否定转移公理的附加观察，与browwer创造的主体反例相一致。
{"title":"Solovay's Relative Consistency Proof for FIM and BI","authors":"J. Moschovakis","doi":"10.1215/00294527-2021-0034","DOIUrl":"https://doi.org/10.1215/00294527-2021-0034","url":null,"abstract":"In 2002 Robert Solovay proved that a subsystem BI of classical second order arithmetic, with bar induction and arithmetical countable choice, can be negatively interpreted in the neutral subsystem BSK of Kleene's intuitionistic analysis FIM using Markov's Principle MP. Combining this result with Kleene's formalized recursive realizability, he established (in primitive recursive arithmetic PRA) that FIM + MP and BI have the same consistency strength. This historical note includes Solovay's original proof, with his permission, and the additional observation that Markov's Principle can be weakened to a double negation shift axiom consistent with Brouwer's creating subject counterexamples.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80481838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1215/00294527-2021-0008
A. Pillay
{"title":"Remarks on Purity of Methods","authors":"A. Pillay","doi":"10.1215/00294527-2021-0008","DOIUrl":"https://doi.org/10.1215/00294527-2021-0008","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74132249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1215/00294527-2021-0002
Graham Leach-Krouse
{"title":"Coalgebra and Abstraction","authors":"Graham Leach-Krouse","doi":"10.1215/00294527-2021-0002","DOIUrl":"https://doi.org/10.1215/00294527-2021-0002","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73232120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1215/00294527-2021-0004
T. McCarthy
This paper is divided into two parts, the first being a point of departure for the second. I will begin by discussing a well-known negative argument due to Mark Lange concerning the explanatory role of mathematical induction. In the first part of the paper, I offer yet another response to Lange’s argument and attempt to characterize the sort of explanatory role played by inductive proofs. That account depends on two structural principles about explanatory proof that look like a fragment of a constructive semantics for that concept. The remainder of the paper fills out this semantics and explores its consequences. It will be clear that this framework does not constitute a fully general characterization of the concept of mathematical proof; the question will be whether there is a natural class of proofs that it does characterize. My answer will be that it nicely describes what I shall call grounding explanatory proofs. A proof of this sort explains the sentence proved in terms of the grounds of the fact that it describes. I will conclude by briefly exploring the connections between grounding proofs and the notion of purity.
{"title":"Induction, Constructivity, and Grounding","authors":"T. McCarthy","doi":"10.1215/00294527-2021-0004","DOIUrl":"https://doi.org/10.1215/00294527-2021-0004","url":null,"abstract":"This paper is divided into two parts, the first being a point of departure for the second. I will begin by discussing a well-known negative argument due to Mark Lange concerning the explanatory role of mathematical induction. In the first part of the paper, I offer yet another response to Lange’s argument and attempt to characterize the sort of explanatory role played by inductive proofs. That account depends on two structural principles about explanatory proof that look like a fragment of a constructive semantics for that concept. The remainder of the paper fills out this semantics and explores its consequences. It will be clear that this framework does not constitute a fully general characterization of the concept of mathematical proof; the question will be whether there is a natural class of proofs that it does characterize. My answer will be that it nicely describes what I shall call grounding explanatory proofs. A proof of this sort explains the sentence proved in terms of the grounds of the fact that it describes. I will conclude by briefly exploring the connections between grounding proofs and the notion of purity.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73019701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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