Results for 'hereditarily finite sets'

999 found
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  1.  22
    A Hierarchy of Hereditarily Finite Sets.Laurence Kirby - 2008 - Archive for Mathematical Logic 47 (2):143-157.
    This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy.
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  2.  20
    Induction and Foundation in the Theory of Hereditarily Finite Sets.Flavio Previale - 1994 - Archive for Mathematical Logic 33 (3):213-241.
    The paper contains an axiomatic treatment of the intuitionistic theory of hereditarily finite sets, based on an induction axiom-schema and a finite set of single axioms. The main feature of the principle of induction used (due to Givant and Tarski) is that it incorporates Foundation. On the analogy of what is done in Arithmetic, in the axiomatic system selected the transitive closure of the membership relation is taken as a primitive notion, so as to permit an (...)
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  3.  11
    A Machine-Assisted Proof of Gödel’s Incompleteness Theorems for the Theory of Hereditarily Finite Sets.Lawrence C. Paulson - 2014 - Review of Symbolic Logic 7 (3):484-498.
  4. Hereditarily Finite Finsler Sets.David Booth - 1990 - Journal of Symbolic Logic 55 (2):700-706.
  5.  23
    Finitary Set Theory.Laurence Kirby - 2009 - Notre Dame Journal of Formal Logic 50 (3):227-244.
    I argue for the use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory. It allows a simplified axiomatization for the first-order theory of hereditarily finite sets based on an induction schema and a rigorous characterization of the primitive recursive set functions. The latter leads to a primitive recursive presentation of arithmetical operations on finite sets.
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  6.  11
    Decidability of ∃*∀∀-Sentences in HF.D. Bellè & F. Parlamento - 2008 - Notre Dame Journal of Formal Logic 49 (1):55-64.
    Let HF be the collection of the hereditarily finite well-founded sets and let the primitive language of set theory be the first-order language which contains binary symbols for equality and membership only. As announced in a previous paper by the authors, "Truth in V for ∃*∀∀-sentences is decidable," truth in HF for ∃*∀∀-sentences of the primitive language is decidable. The paper provides the proof of that claim.
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  7. Rational Choice on Non-Finite Sets by Means of Expansion-Contraction Axioms.M. Carmen Sánchez - 1998 - Theory and Decision 45 (1):1-17.
    The rationalization of a choice function, in terms of assumptions that involve expansion or contraction properties of the feasible set, over non-finite sets is analyzed. Schwartz's results, stated in the finite case, are extended to this more general framework. Moreover, a characterization result when continuity conditions are imposed on the choice function, as well as on the binary relation that rationalizes it, is presented.
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  8.  17
    Substandard Models of Finite Set Theory.Laurence Kirby - 2010 - Mathematical Logic Quarterly 56 (6):631-642.
    A survey of the isomorphic submodels of Vω, the set of hereditarily finite sets. In the usual language of set theory, Vω has 2ℵ0 isomorphic submodels. But other set-theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of Vω.
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  9.  6
    Choices of Convenient Sets.Antonín Sochor - 1994 - Mathematical Logic Quarterly 40 (1):51-60.
    Proceeding in the theory with extensionality, comprehension for classes, existence of the empty set and the assumption the addition of one element to a set makes again a set we show a week assumption which guarantees existence of a saturated elementary extension of the system of hereditarily finite sets.
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  10.  19
    Strongly Adequate Sets and Adding a Club with Finite Conditions.John Krueger - 2014 - Archive for Mathematical Logic 53 (1-2):119-136.
    We continue the study of adequate sets which we began in (Krueger in Forcing with adequate sets of models as side conditions) by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω 2 with finite conditions, thereby showing that a version of the forcing posets of Friedman (Set (...)
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  11.  23
    Self-Embeddings of Computable Trees.Stephen Binns, Bjørn Kjos-Hanssen, Manuel Lerman, James H. Schmerl & Reed Solomon - 2008 - Notre Dame Journal of Formal Logic 49 (1):1-37.
    We divide the class of infinite computable trees into three types. For the first and second types, 0' computes a nontrivial self-embedding while for the third type 0'' computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite Π01 antichain. This result is optimal and has connections (...)
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  12.  19
    On Uniform Definability of Types Over Finite Sets.Vincent Guingona - 2012 - Journal of Symbolic Logic 77 (2):499-514.
    In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.
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  13. An Induction Principle and Pigeonhole Principles for K-Finite Sets.Andreas Blass - 1995 - Journal of Symbolic Logic 60 (4):1186-1193.
    We establish a course-of-values induction principle for K-finite sets in intuitionistic type theory. Using this principle, we prove a pigeonhole principle conjectured by Bénabou and Loiseau. We also comment on some variants of this pigeonhole principle.
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  14.  32
    Grasping Infinity by Finite Sets.Ferrante Formato & Giangiacomo Gerla - 1998 - Mathematical Logic Quarterly 44 (3):383-393.
    We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of wide sets.
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  15.  18
    On Topological Properties of Ultraproducts of Finite Sets.Gábor Sági & Saharon Shelah - 2005 - Mathematical Logic Quarterly 51 (3):254-257.
    In [3] a certain family of topological spaces was introduced on ultraproducts. These spaces have been called ultratopologies and their definition was motivated by model theory of higher order logics. Ultratopologies provide a natural extra topological structure for ultraproducts. Using this extra structure in [3] some preservation and characterization theorems were obtained for higher order logics. The purely topological properties of ultratopologies seem interesting on their own right. We started to study these properties in [2], where some questions remained open. (...)
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  16.  19
    The First-Order Structure of Weakly Dedekind-Finite Sets.A. C. Walczak-Typke - 2005 - Journal of Symbolic Logic 70 (4):1161 - 1170.
    We show that infinite sets whose power-sets are Dedekind-finite can only carry N₀-categorical first order structures. We identify other subclasses of this class of Dedekind-finite sets, and discuss their possible first order structures.
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  17.  28
    On Hereditarily Countable Sets.Thomas Jech - 1982 - Journal of Symbolic Logic 47 (1):43-47.
    It is shown (in ZF) that every hereditarily countable set has rank less than ω 2 , and that if ℵ 1 is singular then there are hereditarily countable sets of all ranks less than ω 2.
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  18.  36
    Lexicographic Additivity for Multi-Attribute Preferences on Finite Sets.Yutaka Nakamura - 1997 - Theory and Decision 42 (1):1-19.
    This paper explores lexicographically additive representations of multi-attribute preferences on finite sets. Lexicographic additivity combines a lexicographic feature with local value tradeoffs. Tradeoff structures are governed by either transitive or nontransitive additive conjoint measurement. Alternatives are locally traded off when they are close enough within threshold associated with a dominant subset of attributes.
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  19.  9
    Finite Sets and Natural Numbers in Intuitionistic TT.Daniel Dzierzgowski - 1996 - Notre Dame Journal of Formal Logic 37 (4):585-601.
    We show how to interpret Heyting's arithmetic in an intuitionistic version of TT, Russell's Simple Theory of Types. We also exhibit properties of finite sets in this theory and compare them with the corresponding properties in classical TT. Finally, we prove that arithmetic can be interpreted in intuitionistic TT, the subsystem of intuitionistic TT involving only three types. The definitions of intuitionistic TT and its finite sets and natural numbers are obtained in a straightforward way from (...)
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  20.  4
    A Remark on Hereditarily Nonparadoxical Sets.Péter Komjáth - 2016 - Archive for Mathematical Logic 55 (1-2):165-175.
    Call a set A⊆R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A \subseteq \mathbb {R}}$$\end{document}paradoxical if there are disjoint A0,A1⊆A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A_0, A_1 \subseteq A}$$\end{document} such that both A0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A_0}$$\end{document} and A1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A_1}$$\end{document} are equidecomposable with A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A}$$\end{document} via countabbly many translations. X⊆R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...)
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  21.  14
    A Note on Weakly Dedekind Finite Sets.Pimpen Vejjajiva & Supakun Panasawatwong - 2014 - Notre Dame Journal of Formal Logic 55 (3):413-417.
    A set $A$ is Dedekind infinite if there is a one-to-one function from $\omega$ into $A$. A set $A$ is weakly Dedekind infinite if there is a function from $A$ onto $\omega$; otherwise $A$ is weakly Dedekind finite. For a set $M$, let $\operatorname{dfin}^{*}$ denote the set of all weakly Dedekind finite subsets of $M$. In this paper, we prove, in Zermelo–Fraenkel set theory, that $|\operatorname{dfin}^{*}|\lt |\mathcal{P}|$ if $\operatorname{dfin}^{*}$ is Dedekind infinite, whereas $|\operatorname{dfin}^{*}|\lt |\mathcal{P}|$ cannot be proved from (...)
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  22.  48
    Quantum Mechanics Over Sets: A Pedagogical Model with Non-Commutative Finite Probability Theory as its Quantum Probability Calculus.David Ellerman - 2017 - Synthese (12):4863-4896.
    This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability (...)
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  23.  14
    Finite Contractions on Infinite Belief Sets.Sven Ove Hansson - 2012 - Studia Logica 100 (5):907-920.
    Contractions on belief sets that have no finite representation cannot be finite in the sense that only a finite number of sentences is removed. However, such contractions can be delimited so that the actual change takes place in a logically isolated, finite-based part of the belief set. A construction that answers to this principle is introduced, and is axiomatically characterized. It turns out to coincide with specified meet contraction.
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  24.  36
    Unique Nontransitive Measurement on Finite Sets.Peter C. Fishburn - 1990 - Theory and Decision 28 (1):21-46.
  25.  25
    Finite Algebras of Relations Are Representable on Finite Sets.H. Andréka, I. Hodkinson & I. Németi - 1999 - Journal of Symbolic Logic 64 (1):243-267.
    Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.
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  26.  25
    Divisibility of Dedekind Finite Sets.David Blair, Andreas Blass & Paul Howard - 2005 - Journal of Mathematical Logic 5 (1):49-85.
    A Dedekind-finite set is said to be divisible by a natural number n if it can be partitioned into pieces of size n. We study several aspects of this notion, as well as the stronger notion of being partitionable into n pieces of equal size. Among our results are that the divisors of a Dedekind-finite set can consistently be any set of natural numbers, that a Dedekind-finite power of 2 cannot be divisible by 3, and that a (...)
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  27.  6
    Finite Algebras of Relations Are Representable on Finite Sets.H. Andreka, I. Hodkinson & I. Nemeti - 1999 - Journal of Symbolic Logic 64 (1):243-267.
    Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.
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  28.  22
    Complete Axiomatisations of Properties of Finite Sets.Thomas Agotnes & Michal Walicki - 2008 - Logic Journal of the IGPL 16 (3):293-313.
    We study a logic whose formulae are interpreted as properties of a finite set over some universe. The language is propositional, with two unary operators inclusion and extension, both taking a finite set as argument. We present a basic Hilbert-style axiomatisation, and study its completeness. The main results are syntactic and semantic characterisations of complete extensions of the logic.
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  29.  28
    Wadge Hierarchy and Veblen Hierarchy Part I: Borel Sets of Finite Rank.J. Duparc - 2001 - Journal of Symbolic Logic 66 (1):56-86.
    We consider Borel sets of finite rank $A \subseteq\Lambda^\omega$ where cardinality of Λ is less than some uncountable regular cardinal K. We obtain a "normal form" of A, by finding a Borel set Ω, such that A and Ω continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base K, under the map which sends every Borel (...)
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  30.  13
    Finite Semiotics: Cognitive Sets, Semiotic Vectors, and Semiosic Oscillation.Cameron Shackell - 2019 - Semiotica 2019 (229):211-235.
    The grounding of semiotics in the finiteness of cognition is extended into constructs and methods for analysis by incorporating the assumption that cognition can be similar within and between agents. After examining and formalizing cognitive similarity as an ontological commitment, the recurrence of cognitive states is examined in terms of a “cognitive set.” In the individual, the cognitive set is seen as evolving under the bidirectional, cyclical determination of thought by the historical environment. At the population level, the distributed “global” (...)
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  31.  38
    Hereditary Undecidability of Some Theories of Finite Structures.Ross Willard - 1994 - Journal of Symbolic Logic 59 (4):1254-1262.
    Using a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms.
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  32.  50
    Finite Support Iteration and Strong Measure Zero Sets.Janusz Pawlikowski - 1990 - Journal of Symbolic Logic 55 (2):674-677.
    Any finite support iteration of posets with precalibre ℵ 1 which has the length of cofinality greater than ω 1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ 1 strong measure zero sets.
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  33.  53
    Finite Powers of Strong Measure Zero Sets.Marion Scheepers - 1999 - Journal of Symbolic Logic 64 (3):1295-1306.
    In a previous paper-[17]-we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove: 1. If a metric space has a (...)
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  34.  25
    Decidability and Finite Axiomatizability of Theories of ℵ0-Categorical Partially Ordered Sets.James H. Schmerl - 1981 - Journal of Symbolic Logic 46 (1):101 - 120.
    Every ℵ 0 -categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ 0 -categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ 0 -categorical partially ordered set not embedding one of them has a decidable theory.
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  35.  32
    V. L. MurskiĬ. Nondiscernible Properties of Finite Systems of Identity Relations. Soviet Mathematics, Vol. 12 , Pp. 183–186. , Pp. 520–522.) - George F. McNulty. The Decision Problem for Equational Bases of Algebras. Annals of Mathematical Logic, Vol. 10 , Pp. 193–259. - George F. McNulty. Undecidable Properties of Finite Sets of Equations. The Journal of Symbolic Logic, Vol. 41 , Pp. 589–604. [REVIEW]S. Burris - 1982 - Journal of Symbolic Logic 47 (4):903-904.
  36.  30
    H. Jerome Keisler. Ultraproducts of Finite Sets. The Journal of Symbolic Logic, Vol. 32 , Pp. 47–57.James R. Geiser - 1970 - Journal of Symbolic Logic 35 (4):586.
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  37.  20
    Constructively Complete Finite Sets.Mark Mandelkern - 1988 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 34 (2):97-103.
  38.  10
    Constructively Complete Finite Sets.Mark Mandelkern - 1988 - Mathematical Logic Quarterly 34 (2):97-103.
  39.  51
    Finite Sets and Frege Structures.John L. Bell - 1999 - Journal of Symbolic Logic 64 (4):1552-1556.
    Call a family F of subsets of a set E inductive if ∅ ∈ F and F is closed under unions with disjoint singletons, that is, if ∀X∈F ∀x∈E–X(X ∪ {x} ∈ F]. A Frege structure is a pair (E.
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  40.  30
    Undecidable Properties of Finite Sets of Equations.George F. McNulty - 1976 - Journal of Symbolic Logic 41 (3):589-604.
  41.  6
    Finite sets and infinite sets in weak intuitionistic arithmetic.Takako Nemoto - 2020 - Archive for Mathematical Logic 59 (5):607-657.
    In this paper, we consider, for a set \ of natural numbers, the following notions of finitenessFIN1:There are a natural number l and a bijection f between \\);FIN5:It is not the case that \\), and infinitenessINF1:There are not a natural number l and a bijection f between \\);INF5:\\). In this paper, we systematically compare them in the method of constructive reverse mathematics. We show that the equivalence among them can be characterized by various combinations of induction axioms and non-constructive principles, (...)
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  42.  24
    Ideals of Generalized Finite Sets in Lattices of Α-Recursively Enumerable Sets.Manuel Lerman - 1976 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 22 (1):347-352.
  43.  40
    On the Cardinality of Ultraproduct of Finite Sets.Saharon Shelah - 1970 - Journal of Symbolic Logic 35 (1):83-84.
  44.  13
    Saharon Shelah. On the Cardinality of Ultraproduct of Finite Sets. The Journal of Symbolic Logic, Vol. 35 , Pp. 83–84.A. Slomson - 1973 - Journal of Symbolic Logic 38 (4):650.
  45.  35
    Ultraproducts of Finite Sets.H. Jerome Keisler - 1967 - Journal of Symbolic Logic 32 (1):47-57.
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  46.  15
    Axiom of Choice for Finite Sets.Andrzej Mostowski - 1948 - Journal of Symbolic Logic 13 (1):45-46.
  47.  13
    Markov's Principle, Isols and Dedekind Finite Sets.Charles McCarty - 1988 - Journal of Symbolic Logic 53 (4):1042-1069.
  48. Belief Structures, Possibility Theory and Decomposable Confidence Measures on Finite Sets.Didier Dubois - 1986 - Computers and Artificial Intelligence (5):403--416.
     
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  49.  8
    Ideals of Generalized Finite Sets in Lattices of Α‐Recursively Enumerable Sets.Manuel Lerman - 1976 - Mathematical Logic Quarterly 22 (1):347-352.
  50.  29
    Wiśniewski K.. Weakened Forms of the Axiom of Choice for Finite Sets. Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques Et Physiques, Vol. 16 , Pp. 615–620. [REVIEW]Azriel Lévy - 1971 - Journal of Symbolic Logic 36 (3):543-543.
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