Pub Date : 2022-11-28DOI: 10.4467/20842589rm.22.004.16661
D. Chompitaki, Manos N. Kamarianakis, T. Pheidas
Let pbe a prime number, Fp a finite field with pelements, Fan algebraic extension of Fp and z a variable. We consider the structure of addition and the Frobenius map (i.e., x →xp) in the polynomial rings F[z] and in fields F(z) of rational functions. We prove that any question about F[z] in the structure of addition and Frobenius map may be effectively reduced to questions about the similar structure of the field F. Furthermore, we provide an example which shows that a fact which is true for addition and the Frobenius map in the polynomial rings F[z] fails to be true in F(z). As a consequence, certain methods used to prove model completeness for polynomials do not suffice to prove model completeness for similar structures for fields of rational functions F(z), a problem that remains open even for F= Fp.
{"title":"Notes on the decidability of addition and the Frobenius map for polynomials and rational functions","authors":"D. Chompitaki, Manos N. Kamarianakis, T. Pheidas","doi":"10.4467/20842589rm.22.004.16661","DOIUrl":"https://doi.org/10.4467/20842589rm.22.004.16661","url":null,"abstract":"Let pbe a prime number, Fp a finite field with pelements, Fan algebraic extension of Fp and z a variable. We consider the structure of addition and the Frobenius map (i.e., x →xp) in the polynomial rings F[z] and in fields F(z) of rational functions. We prove that any question about F[z] in the structure of addition and Frobenius map may be effectively reduced to questions about the similar structure of the field F. Furthermore, we provide an example which shows that a fact which is true for addition and the Frobenius map in the polynomial rings F[z] fails to be true in F(z). As a consequence, certain methods used to prove model completeness for polynomials do not suffice to prove model completeness for similar structures for fields of rational functions F(z), a problem that remains open even for F= Fp.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"73 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116980939","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}
Pub Date : 2022-11-28DOI: 10.4467/20842589rm.22.003.16660
Roberta Bonacina, Daniel Misselbeck-Wessel
Menger's graph theorem equates the minimum size of a separating set for non-adjacent vertices a and b with the maximum number of disjoint paths between a and b. By capturing separating sets as models of an entailment relation, we take a formal approach to Menger's result. Upon showing that inconsistency is characterised by the existence of suficiently many disjoint paths, we recover Menger's theorem by way of completeness.
{"title":"A formal approach to Menger's theorem","authors":"Roberta Bonacina, Daniel Misselbeck-Wessel","doi":"10.4467/20842589rm.22.003.16660","DOIUrl":"https://doi.org/10.4467/20842589rm.22.003.16660","url":null,"abstract":"Menger's graph theorem equates the minimum size of a separating set for non-adjacent vertices a and b with the maximum number of disjoint paths between a and b. By capturing separating sets as models of an entailment relation, we take a formal approach to Menger's result. Upon showing that inconsistency is characterised by the existence of suficiently many disjoint paths, we recover Menger's theorem by way of completeness.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124468183","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}
Pub Date : 2022-11-28DOI: 10.4467/20842589RM.22.001.16658
Matthias Eberl
We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely extensible finite. The main adoption is the interpretation of the universal quantifier, which has an implicit reection principle. Each universal quantification refers to an indefinitely large, but finite set. The quantified sets may increase, so after a reference by quantification, a further reference typically uses a larger, still finite set. We present the concepts for classical first-order logic and show that these dynamic models are sound and complete with respect to the usual inference rules. Moreover, a finite set of formulas requires a finite part of the increasing model for a correct interpretation.
{"title":"A Model Theory for the Potential Infinite","authors":"Matthias Eberl","doi":"10.4467/20842589RM.22.001.16658","DOIUrl":"https://doi.org/10.4467/20842589RM.22.001.16658","url":null,"abstract":"We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely extensible finite. The main adoption is the interpretation of the universal quantifier, which has an implicit reection principle. Each universal quantification refers to an indefinitely large, but finite set. The quantified sets may increase, so after a reference by quantification, a further reference typically uses a larger, still finite set. We present the concepts for classical first-order logic and show that these dynamic models are sound and complete with respect to the usual inference rules. Moreover, a finite set of formulas requires a finite part of the increasing model for a correct interpretation.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123498509","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}
Pub Date : 2022-11-28DOI: 10.4467/20842589rm.22.002.16659
P. Lipparini
Contrary to the finitary case, the set Γ(A) of all the non-generators of an infinitary algebra A is not necessarily a subalgebra of A. We show that the phenomenon is ubiquitous: every algebra with at least one infinitary operation can be embedded into some algebra B such that Γ(B) is not a subalgebra of B. As far as expansions are concerned, there are examples of infinite algebras A such that in every expansion B of A the set Γ(B) is a subalgebra of B. However, under relatively weak assumptions on A, it is possible to get some expansion B of A such that Γ(B) fails to be a subalgebra of B.
{"title":"Non-generators in extensions of infinitary algebras","authors":"P. Lipparini","doi":"10.4467/20842589rm.22.002.16659","DOIUrl":"https://doi.org/10.4467/20842589rm.22.002.16659","url":null,"abstract":"Contrary to the finitary case, the set Γ(A) of all the non-generators of an infinitary algebra A is not necessarily a subalgebra of A. We show that the phenomenon is ubiquitous: every algebra with at least one infinitary operation can be embedded into some algebra B such that Γ(B) is not a subalgebra of B. As far as expansions are concerned, there are examples of infinite algebras A such that in every expansion B of A the set Γ(B) is a subalgebra of B. However, under relatively weak assumptions on A, it is possible to get some expansion B of A such that Γ(B) fails to be a subalgebra of B.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128230940","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}
Pub Date : 2022-11-28DOI: 10.4467/20842589rm.22.005.16662
N. Ackerman, M. Karker
In this paper we prove a Lindström like theorem for the logic consisting of arbitrary Boolean combinations of first order sentences. Specifically we show the logic obtained by taking arbitrary, possibly infinite, Boolean combinations of first order sentences in countable languages is the unique maximal abstract logic which is closed under finitary Boolean operations, has occurrence number ω1, has the downward Lüowenheim-Skolem property to ωand the upward Lüowenheim-Skolem property to uncountability, and contains all complete first order theories in countable languages as sentences of the abstract logic. We will also show a similar result holds in the continuous logic framework of [5], i.e. we prove a Lindström like theorem for the abstract continuous logic consisting of Boolean combinations of first order closed conditions. Specifically we show the abstract continuous logic consisting of arbitrary Boolean combinations of closed conditions is the unique maximal abstract continuous logic which is closed under approximate isomorphisms on countable structures, is closed under finitary Boolean operations, has occurrence number ω1, has the downward Lüowenheim-Skolem property toω, the upward Lüowenheim-Skolem property to uncountability and contains all first order theories in countable languages as sentences of the abstract logic.
{"title":"A Maximality Theorem for Continuous First Order Theories","authors":"N. Ackerman, M. Karker","doi":"10.4467/20842589rm.22.005.16662","DOIUrl":"https://doi.org/10.4467/20842589rm.22.005.16662","url":null,"abstract":"In this paper we prove a Lindström like theorem for the logic consisting of arbitrary Boolean combinations of first order sentences. Specifically we show the logic obtained by taking arbitrary, possibly infinite, Boolean combinations of first order sentences in countable languages is the unique maximal abstract logic which is closed under finitary Boolean operations, has occurrence number ω1, has the downward Lüowenheim-Skolem property to ωand the upward Lüowenheim-Skolem property to uncountability, and contains all complete first order theories in countable languages as sentences of the abstract logic. We will also show a similar result holds in the continuous logic framework of [5], i.e. we prove a Lindström like theorem for the abstract continuous logic consisting of Boolean combinations of first order closed conditions. Specifically we show the abstract continuous logic consisting of arbitrary Boolean combinations of closed conditions is the unique maximal abstract continuous logic which is closed under approximate isomorphisms on countable structures, is closed under finitary Boolean operations, has occurrence number ω1, has the downward Lüowenheim-Skolem property toω, the upward Lüowenheim-Skolem property to uncountability and contains all first order theories in countable languages as sentences of the abstract logic.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129546292","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}
Pub Date : 2021-11-10DOI: 10.4467/20842589rm.21.004.14376
Satoru Niki, Hitoshi Omori
We investigate an expansion of positive intuitionistic logic obtained by adding a constant Ω introduced by Lloyd Humberstone. Our main results include a sound and strongly complete axiomatization, some comparisons to other expansions of intuitionistic logic obtained by adding actuality and empirical negation, and an algebraic semantics. We also brie y discuss its connection to classical logic.
{"title":"A note on Humberstone's constant Ω","authors":"Satoru Niki, Hitoshi Omori","doi":"10.4467/20842589rm.21.004.14376","DOIUrl":"https://doi.org/10.4467/20842589rm.21.004.14376","url":null,"abstract":"We investigate an expansion of positive intuitionistic logic obtained by adding a constant Ω introduced by Lloyd Humberstone. Our main results include a sound and strongly complete axiomatization, some comparisons to other expansions of intuitionistic logic obtained by adding actuality and empirical negation, and an algebraic semantics. We also brie y discuss its connection to classical logic.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124013784","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}
Pub Date : 2021-11-10DOI: 10.4467/20842589rm.21.003.14375
Tin Perkov, Luka Mikec
We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.
{"title":"Tableau-based translation from first-order logic to modal logic","authors":"Tin Perkov, Luka Mikec","doi":"10.4467/20842589rm.21.003.14375","DOIUrl":"https://doi.org/10.4467/20842589rm.21.003.14375","url":null,"abstract":"We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116466969","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}
Pub Date : 2021-11-10DOI: 10.4467/20842589rm.21.002.14374
Conrado Gomez, M. Marcos, H. J. S. Martín
The aim of this paper is to investigate the relation between the strong and the "weak" or intuitionistic negation in Nelson algebras. To do this, we define the variety of Kleene algebras with intuitionistic negation and explore the Kalman's construction for pseudocomplemented distributive lattices. We also study the centered algebras of this variety.
{"title":"On the relation of negations in Nelson algebras","authors":"Conrado Gomez, M. Marcos, H. J. S. Martín","doi":"10.4467/20842589rm.21.002.14374","DOIUrl":"https://doi.org/10.4467/20842589rm.21.002.14374","url":null,"abstract":"The aim of this paper is to investigate the relation between the strong and the \"weak\" or intuitionistic negation in Nelson algebras. To do this, we define the variety of Kleene algebras with intuitionistic negation and explore the Kalman's construction for pseudocomplemented distributive lattices. We also study the centered algebras of this variety.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134111026","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}
Pub Date : 2021-11-09DOI: 10.4467/20842589rm.21.001.14373
Nattapon Sonpanow, P. Vejjajiva
Some cardinal characteristics related to the covering number and the uniformity of the meagre ideal
一些基本特征与覆盖数和理想的均匀性有关
{"title":"Some cardinal characteristics related to the covering number and the uniformity of the meagre ideal","authors":"Nattapon Sonpanow, P. Vejjajiva","doi":"10.4467/20842589rm.21.001.14373","DOIUrl":"https://doi.org/10.4467/20842589rm.21.001.14373","url":null,"abstract":"Some cardinal characteristics related to the covering number and the uniformity of the meagre ideal","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"266 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117044556","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}
Pub Date : 2021-05-21DOI: 10.4467/20842589rm.21.005.14377
R. A. Freire
We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete car- dinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure theorem for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.
{"title":"Embeddability Between Orderings and GCH","authors":"R. A. Freire","doi":"10.4467/20842589rm.21.005.14377","DOIUrl":"https://doi.org/10.4467/20842589rm.21.005.14377","url":null,"abstract":"We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete car- dinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure theorem for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.","PeriodicalId":447333,"journal":{"name":"Reports Math. Log.","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114665771","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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