Pub Date : 2019-12-28DOI: 10.1215/00294527-2021-0028
G. Shen
For a cardinal $mathfrak{a}$, let $mathrm{fin}(mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $mathfrak{a}$ and all natural numbers $n$, [ 2^{mathrm{fin}(mathfrak{a})^n}=2^{[mathrm{fin}(mathfrak{a})]^n}. ] On the other hand, it is proved that the following statement is consistent with $mathsf{ZF}$: there exists an infinite cardinal $mathfrak{a}$ such that [ 2^{mathrm{fin}(mathfrak{a})}<2^{mathrm{fin}(mathfrak{a})^2}<2^{mathrm{fin}(mathfrak{a})^3}
{"title":"A Choice-Free Cardinal Equality","authors":"G. Shen","doi":"10.1215/00294527-2021-0028","DOIUrl":"https://doi.org/10.1215/00294527-2021-0028","url":null,"abstract":"For a cardinal $mathfrak{a}$, let $mathrm{fin}(mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $mathfrak{a}$ and all natural numbers $n$, [ 2^{mathrm{fin}(mathfrak{a})^n}=2^{[mathrm{fin}(mathfrak{a})]^n}. ] On the other hand, it is proved that the following statement is consistent with $mathsf{ZF}$: there exists an infinite cardinal $mathfrak{a}$ such that [ 2^{mathrm{fin}(mathfrak{a})}<2^{mathrm{fin}(mathfrak{a})^2}<2^{mathrm{fin}(mathfrak{a})^3}<dots<2^{mathrm{fin}(mathrm{fin}(mathfrak{a}))}. ]","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72730335","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 : 2019-11-01DOI: 10.1215/00294527-2019-0026
Erik Walsberg
We give a short proof of the Marker-Steinhorn theorem for ominimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.
{"title":"The Marker-Steinhorn Theorem via Definable Linear Orders","authors":"Erik Walsberg","doi":"10.1215/00294527-2019-0026","DOIUrl":"https://doi.org/10.1215/00294527-2019-0026","url":null,"abstract":"We give a short proof of the Marker-Steinhorn theorem for ominimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78714899","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 : 2019-11-01DOI: 10.1215/00294527-2019-0021
H. Mildenberger, S. Shelah
{"title":"Specializing Aronszajn Trees with Strong Axiom A and Halving","authors":"H. Mildenberger, S. Shelah","doi":"10.1215/00294527-2019-0021","DOIUrl":"https://doi.org/10.1215/00294527-2019-0021","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91361312","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 : 2019-11-01DOI: 10.1215/00294527-2019-0024
D. Asperó, Tapani Hyttinen, V. Kulikov, Miguel Moreno
Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal kappa. We show the consistency of E^{lambda^{++},lambda^{++}}_{lambdatext{-club}}, the relation of equivalence modulo the non-stationary ideal restricted to S^{lambda^{++}}_lambda in the space (lambda^{++})^{lambda^{++}}, being continuously reducible to E^{2,lambda^{++}}_{lambda^+text{-club}}, the relation of equivalence modulo the non-stationary ideal restricted to S^{lambda^{++}}_{lambda^+} in the space 2^{lambda^{++}}. Then we show that for kappa ineffable E^{2, kappa}_{text{reg}}, the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space 2^{kappa}, is Sigma^1_1-complete. We finish by showing, for Pi_2^1-indescribable kappa, that the isomorphism relation between dense linear orders of cardinality kappa is Sigma^1_1-complete.
{"title":"Reducibility of Equivalence Relations Arising from Nonstationary Ideals under Large Cardinal Assumptions","authors":"D. Asperó, Tapani Hyttinen, V. Kulikov, Miguel Moreno","doi":"10.1215/00294527-2019-0024","DOIUrl":"https://doi.org/10.1215/00294527-2019-0024","url":null,"abstract":"Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal kappa. We show the consistency of E^{lambda^{++},lambda^{++}}_{lambdatext{-club}}, the relation of equivalence modulo the non-stationary ideal restricted to S^{lambda^{++}}_lambda in the space (lambda^{++})^{lambda^{++}}, being continuously reducible to E^{2,lambda^{++}}_{lambda^+text{-club}}, the relation of equivalence modulo the non-stationary ideal restricted to S^{lambda^{++}}_{lambda^+} in the space 2^{lambda^{++}}. Then we show that for kappa ineffable E^{2, kappa}_{text{reg}}, the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space 2^{kappa}, is Sigma^1_1-complete. We finish by showing, for Pi_2^1-indescribable kappa, that the isomorphism relation between dense linear orders of cardinality kappa is Sigma^1_1-complete.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77607752","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 : 2019-11-01DOI: 10.1215/00294527-2019-0028
K. Ng, Hongyuan Yu
We study the degree structure of the ω-c.e., n-c.e. and Π1 equivalence relations under the computable many-one reducibility. In particular we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-c.e. equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e. and Π1 equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.
{"title":"On the Degree Structure of Equivalence Relations Under Computable Reducibility","authors":"K. Ng, Hongyuan Yu","doi":"10.1215/00294527-2019-0028","DOIUrl":"https://doi.org/10.1215/00294527-2019-0028","url":null,"abstract":"We study the degree structure of the ω-c.e., n-c.e. and Π1 equivalence relations under the computable many-one reducibility. In particular we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-c.e. equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e. and Π1 equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86669535","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 : 2019-10-16DOI: 10.1215/00294527-2022-0001
Sam Sanders
The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.
{"title":"Representations and the Foundations of Mathematics","authors":"Sam Sanders","doi":"10.1215/00294527-2022-0001","DOIUrl":"https://doi.org/10.1215/00294527-2022-0001","url":null,"abstract":"The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44287071","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 : 2019-09-30DOI: 10.1215/00294527-2019-0022
Bruno Bentzen
This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book.
{"title":"Frege on Referentiality and Julius Caesar in Grundgesetze Section 10","authors":"Bruno Bentzen","doi":"10.1215/00294527-2019-0022","DOIUrl":"https://doi.org/10.1215/00294527-2019-0022","url":null,"abstract":"This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the \"problem of indeterminacy of reference\" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84955561","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 : 2019-08-01DOI: 10.1215/00294527-2019-0008
Brian Hill, F. Poggiolesi
The goal of this paper is to take a step towards the resolution of the problem of finding an analytic sequent calculus for the logic of proofs. For this, we focus on the system Ilp, the intuitionistic version of the logic of proofs. First we present the sequent calculus Gilp that is sound and complete with respect to the system Ilp; we prove that Gilp is cut-free and contraction-free, but it still does not enjoy the subformula property. Then, we enrich the language of the logic of proofs and we formulate in this language a second Gentzen calculus Gilp∗. We show that Gilp∗ is a conservative extension of Gilp, and that Gilp∗ satisfies the subformula property. Keyword cut-elimination, logic of proofs, normalisation, proof sequents 2010 MSC: 03F05, 03B60
{"title":"An Analytic Calculus for the Intuitionistic Logic of Proofs","authors":"Brian Hill, F. Poggiolesi","doi":"10.1215/00294527-2019-0008","DOIUrl":"https://doi.org/10.1215/00294527-2019-0008","url":null,"abstract":"The goal of this paper is to take a step towards the resolution of the problem of finding an analytic sequent calculus for the logic of proofs. For this, we focus on the system Ilp, the intuitionistic version of the logic of proofs. First we present the sequent calculus Gilp that is sound and complete with respect to the system Ilp; we prove that Gilp is cut-free and contraction-free, but it still does not enjoy the subformula property. Then, we enrich the language of the logic of proofs and we formulate in this language a second Gentzen calculus Gilp∗. We show that Gilp∗ is a conservative extension of Gilp, and that Gilp∗ satisfies the subformula property. Keyword cut-elimination, logic of proofs, normalisation, proof sequents 2010 MSC: 03F05, 03B60","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74225581","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 : 2019-08-01DOI: 10.1215/00294527-2019-0010
Jie Fan
Inspired by Hintikka's treatment of question embedding verbs in [8] and the variations of noncontingency operator, we propose a logic with strong noncontingency operator $blacktriangle$ as the only primitive modality. A proposition is strongly noncontingent, if no matter whether it is true or false, it does it necessarily; otherwise, it is weakly contingent. This logic is not a normal modal logic, since $blacktriangle(phitopsi)to(blacktrianglephitoblacktrianglepsi)$ is invalid. We compare the relative expressivity of this logic and other logics, such as standard modal logic, noncontingency logic, and logic of essence and accident, and investigate its frame definability. Apart from these results, we also propose a suitable notion of bisimulation for the logic of strong noncontingency, based on which we characterize this logic within modal logic and within first-order logic. We also axiomatize the logic of strong noncontingency over various frame classes. Our work is also related to the treatment of agreement operator in [10].
{"title":"Strong Noncontingency: On the Modal Logics of an Operator Expressively Weaker Than Necessity","authors":"Jie Fan","doi":"10.1215/00294527-2019-0010","DOIUrl":"https://doi.org/10.1215/00294527-2019-0010","url":null,"abstract":"Inspired by Hintikka's treatment of question embedding verbs in [8] and the variations of noncontingency operator, we propose a logic with strong noncontingency operator $blacktriangle$ as the only primitive modality. A proposition is strongly noncontingent, if no matter whether it is true or false, it does it necessarily; otherwise, it is weakly contingent. This logic is not a normal modal logic, since $blacktriangle(phitopsi)to(blacktrianglephitoblacktrianglepsi)$ is invalid. We compare the relative expressivity of this logic and other logics, such as standard modal logic, noncontingency logic, and logic of essence and accident, and investigate its frame definability. Apart from these results, we also propose a suitable notion of bisimulation for the logic of strong noncontingency, based on which we characterize this logic within modal logic and within first-order logic. We also axiomatize the logic of strong noncontingency over various frame classes. Our work is also related to the treatment of agreement operator in [10].","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73461021","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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