Pub Date : 2008-05-29DOI: 10.1017/CBO9780511910616.004
A. Enayat, J. Schmerl, A. Visser
Finite set theory, here denoted ZFfin, is the theory ob- tained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An !-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) em- ployed the Bernays-Rieger method of permutations to construct a recursive !-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets V!). In this paper we initiate the metamathematical investigation of !-models of ZFfin. In par- ticular, we present a new method for building !-models of ZFfin that leads to a perspicuous construction of recursive nonstandard !-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an !-model. The central theorem of the paper is the following:
{"title":"ω-Models of finite set theory","authors":"A. Enayat, J. Schmerl, A. Visser","doi":"10.1017/CBO9780511910616.004","DOIUrl":"https://doi.org/10.1017/CBO9780511910616.004","url":null,"abstract":"Finite set theory, here denoted ZFfin, is the theory ob- tained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An !-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) em- ployed the Bernays-Rieger method of permutations to construct a recursive !-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets V!). In this paper we initiate the metamathematical investigation of !-models of ZFfin. In par- ticular, we present a new method for building !-models of ZFfin that leads to a perspicuous construction of recursive nonstandard !-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an !-model. The central theorem of the paper is the following:","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"29 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125920177","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 : 2006-01-15DOI: 10.1002/0470018860.S00231
M. Moortgat
Categorial grammar is a lexicalized grammar formalism based on logical type theory. A categorial lexicon assigns one or more types to the atomic elements of a language; the assembly of form and meaning is accounted for in terms of the rules of inference for these types, seen as formulae of a grammar logic. Cross-linguistic variation results from extending the invariant core of the grammar logic with facilities for structural reasoning. Keywords: categories; types; processing; parsing; deduction
{"title":"Categorial Grammar and Formal Semantics","authors":"M. Moortgat","doi":"10.1002/0470018860.S00231","DOIUrl":"https://doi.org/10.1002/0470018860.S00231","url":null,"abstract":"Categorial grammar is a lexicalized grammar formalism based on logical type theory. A categorial lexicon assigns one or more types to the atomic elements of a language; the assembly of form and meaning is accounted for in terms of the rules of inference for these types, seen as formulae of a grammar logic. Cross-linguistic variation results from extending the invariant core of the grammar logic with facilities for structural reasoning. \u0000 \u0000 \u0000Keywords: \u0000 \u0000categories; \u0000types; \u0000processing; \u0000parsing; \u0000deduction","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128835402","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 : 2004-04-01DOI: 10.1017/9781316755747.019
A. Visser
In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, bi-interpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under such-and-such conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation. We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free’ explication of the notion of axiom scheme. Also we give a closer analysis of the object-language/ meta-language distinction. Our basic category can be enriched with a form of 2-structure. We use this 2-structure to characterize a salient subclass of interpetations, the direct interpretations, and we use the 2-structure to characterize induction. Using this last characterization, we prove a theorem that has as a consequence that, if two extensions of Peano Arithmetic in the arithmetical language are synonymous, then they are identical. Finally, we study preservation of properties over certain morphisms.
{"title":"Categories of theories and interpretations","authors":"A. Visser","doi":"10.1017/9781316755747.019","DOIUrl":"https://doi.org/10.1017/9781316755747.019","url":null,"abstract":"In this paper we study categories of theories and interpretations. In \u0000these categories, notions of sameness of theories, like synonymy, bi-interpretability \u0000and mutual interpretability, take the form of isomorphism. \u0000We study the usual notions like monomorphism and product in the \u0000various theories. We provide some examples to separate notions across \u0000categories. In contrast, we show that, in some cases, notions in different \u0000categories do coincide. E.g., we can, under such-and-such conditions, infer \u0000synonymity of two theories from their being equivalent in the sense of a \u0000coarser equivalence relation. \u0000We illustrate that the categories offer an appropriate framework for \u0000conceptual analysis of notions. For example, we provide a ‘coordinate \u0000free’ explication of the notion of axiom scheme. Also we give a closer \u0000analysis of the object-language/ meta-language distinction. \u0000Our basic category can be enriched with a form of 2-structure. We use \u0000this 2-structure to characterize a salient subclass of interpetations, the direct \u0000interpretations, and we use the 2-structure to characterize induction. \u0000Using this last characterization, we prove a theorem that has as a consequence \u0000that, if two extensions of Peano Arithmetic in the arithmetical \u0000language are synonymous, then they are identical. \u0000Finally, we study preservation of properties over certain morphisms.","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123499174","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 : 2003-03-01DOI: 10.1017/9781316755723.005
L. Beklemishev
In [6] an approach to proof-theoretic analysis of Peano arithmetic based an the motion of graded provability algebra was suggested. Here we present a provability-algebraic version of the independent combinatorial Hydra battle principle. This allows for simple independence proofs of both principles based on provability-algebraic methods.
{"title":"The Worm principle","authors":"L. Beklemishev","doi":"10.1017/9781316755723.005","DOIUrl":"https://doi.org/10.1017/9781316755723.005","url":null,"abstract":"In [6] an approach to proof-theoretic analysis of Peano arithmetic \u0000based an the motion of graded provability algebra was suggested. Here we present a provability-algebraic version of the independent combinatorial Hydra battle principle. This allows for simple independence proofs of both principles based on provability-algebraic methods.","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"219 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129438158","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}
Among those addressed by Putnam’s model-theoretic argument it is common opinion that the argument is invalid because question-begging. If the standard analysis of the argument is along the right lines, then what has been called the ‘just more theory move’ is to be held responsible for this. In the present paper, an alternative reading of Putnam’s argument is offered that makes the ‘just more theory move’ come out perfectly legitimate, and the argument as a whole valid.
{"title":"Putnam’s Model-Theoretic Argument Reconstructed","authors":"I. Douven","doi":"10.2307/2564709","DOIUrl":"https://doi.org/10.2307/2564709","url":null,"abstract":"Among those addressed by Putnam’s model-theoretic argument it is common opinion that the argument is invalid because question-begging. If the standard analysis of the argument is along the right lines, then what has been called the ‘just more theory move’ is to be held responsible for this. In the present paper, an alternative reading of Putnam’s argument is offered that makes the ‘just more theory move’ come out perfectly legitimate, and the argument as a whole valid.","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115583972","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 : 1999-05-01DOI: 10.1007/978-94-011-4574-9_15
M. Moortgat
{"title":"Labelled deduction in the composition of form and meaning","authors":"M. Moortgat","doi":"10.1007/978-94-011-4574-9_15","DOIUrl":"https://doi.org/10.1007/978-94-011-4574-9_15","url":null,"abstract":"","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"194 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115235042","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 : 1996-03-13DOI: 10.1111/J.1755-2567.1997.TB00748.X
V. Shavrukov
For a formal theory T the diagonalizable algebra a k a Magari algebra of T denoted DT is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T the equivalence class of a sentence is sent by T to the equivalence class of the T sentence expressing that T proves It was shown in Shavrukov that the diagonalizable algebras of PA and ZF as well as the diagonalizable algebras of similarly related pairs of sound theories are not isomorphic Neither are these algebras rst order equivalent Shavrukov Theorem In the present paper we establish a su cient condition which we name B co herence for the diagonalizable algebras of two theories to be isomorphic It is then immediately seen that DZF DGB which answers a question of Smory nski We also construct non identity automorphisms of diagonalizable algebras of all theories un der consideration The techniques we use are a combination of those developed in the context of partially conservative sentences cf Lindstr om and those of Pour El Kripke A related construction appears in Solovay Theorem
正式理论T对角化的代数k T的Magari代数表示DT是T的Lindenbaum句子代数具有一元运算符T起源于只是谓词(T)发送一个句子的等价类T等价类的句子表达T证明Shavrukov所示,PA的对角化的代数和ZF以及类似的对角化的代数相关的对声音不同构理论没有这些代数rst秩序等价Shavrukov定理在本文我们建立一个苏字母系数条件我们名字的对角化的代数B公司以后连同两个理论同构然后立即看到DZF DGB这答案的问题Smory nski我们也非身份对角化的代数的同构理论联合国der考虑我们使用的技术是一个组合的环境中开发的部分linstrom的保守句和Pour El Kripke的保守句在索洛维定理中出现了相关的结构
{"title":"Isomorphisms of diagonalizable algebras","authors":"V. Shavrukov","doi":"10.1111/J.1755-2567.1997.TB00748.X","DOIUrl":"https://doi.org/10.1111/J.1755-2567.1997.TB00748.X","url":null,"abstract":"For a formal theory T the diagonalizable algebra a k a Magari algebra of T denoted DT is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T the equivalence class of a sentence is sent by T to the equivalence class of the T sentence expressing that T proves It was shown in Shavrukov that the diagonalizable algebras of PA and ZF as well as the diagonalizable algebras of similarly related pairs of sound theories are not isomorphic Neither are these algebras rst order equivalent Shavrukov Theorem In the present paper we establish a su cient condition which we name B co herence for the diagonalizable algebras of two theories to be isomorphic It is then immediately seen that DZF DGB which answers a question of Smory nski We also construct non identity automorphisms of diagonalizable algebras of all theories un der consideration The techniques we use are a combination of those developed in the context of partially conservative sentences cf Lindstr om and those of Pour El Kripke A related construction appears in Solovay Theorem","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"158 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1996-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130614672","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 : 1986-04-01DOI: 10.1017/CBO9780511608841.012
C. Koymans, J. Mulder
A version of the Alternating Bit Protocol is verified by means of Process Algebra. To avoid a combinatorial explosion, a notion of "modules" is introduced and the protocol is divided in two such modules. A method is developed for verifying conglomerates of modules and applied to the motivating example.
{"title":"A modular approach to protocol verification using process algebra","authors":"C. Koymans, J. Mulder","doi":"10.1017/CBO9780511608841.012","DOIUrl":"https://doi.org/10.1017/CBO9780511608841.012","url":null,"abstract":"A version of the Alternating Bit Protocol is verified by means of \u0000Process Algebra. To avoid a combinatorial explosion, a notion of \"modules\" \u0000is introduced and the protocol is divided in two such modules. A method \u0000is developed for verifying conglomerates of modules and applied to the \u0000motivating example.","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1986-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123254626","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 : 1900-01-01DOI: 10.1007/978-94-017-9673-6_16
A. Enayat, A. Visser
{"title":"New Constructions of Satisfaction Classes","authors":"A. Enayat, A. Visser","doi":"10.1007/978-94-017-9673-6_16","DOIUrl":"https://doi.org/10.1007/978-94-017-9673-6_16","url":null,"abstract":"","PeriodicalId":161799,"journal":{"name":"Logic group preprint series","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117261814","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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