On Vopenka's Ultrafinitism
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Biequivalence Vector Spaces in the Alternative Set Theory
Comment.Math.Univ.Carolin. 32,3 (1991)517–544 517 Biequivalence vector spaces in the alternative set theory Miroslav Smˇ ´ıd, Pavol Zlatoˇs Abstract. As a counterpart to classical topological vector spaces in the alternative set the- ory, biequivalence vector spaces (over the field Q of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view to- wards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total convexity of the monad and/or of the galaxy of 0. Finally, the existence of a rather strong type of basis for a fairly extensive area of biequivalence vector spaces, containing all the most important particular cases, is established. Keywords: alternative set theory, biequivalence, vector space, monad, galaxy, symmetric Sd-closure, dual, valuation, norm, convex, basis Classification: Primary 46Q05, 46A06, 46A35; Secondary 03E70, 03H05, 46A09 Contents: 0. Introduction 1. Notation and preliminaries 2. Symmetric Sd-closures 3. Vector spaces over Q 4. Biequivalence vector spaces 5. Duals 6. Valuations on vector spaces 7. The envelope operation 8. Bases in biequivalence vector spaces 0. Introduction. The aim of this paper is to lay a foundation to the investigation of topological (or perhaps also bornological) vector spaces within the framework of the alternative set theory (AST), which could enable a rather elementary exposition of some topics of functional analysis reducing them to the study of formally finite dimensional vector spaces equipped with some additional “nonsharp” or “hazy” first order structure representing the topology. -
Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
An Outline of Algebraic Set Theory
An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16]. -
Self-Similarity in the Foundations
Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory . -
On Lattices and Their Ideal Lattices, and Posets and Their Ideal Posets
This is the final preprint version of a paper which appeared at Tbilisi Math. J. 1 (2008) 89-103. Published version accessible to subscribers at http://www.tcms.org.ge/Journals/TMJ/Volume1/Xpapers/tmj1_6.pdf On lattices and their ideal lattices, and posets and their ideal posets George M. Bergman 1 ∗ 1 Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA E-mail: [email protected] Abstract For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P; including the empty set, and let id(P ) = Id(P )−f?g: This note obtains various results to the effect that Id(P ) is always, and id(P ) often, \essentially larger" than P: In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone map) from Id(P ) into P; and, going the other way, that an upper semilattice P admits no semilattice homomorphism from any subsemilattice of itself onto Id(P ): The slightly smaller object id(P ) is known to be isomorphic to P if and only if P has ascending chain condition. This result is strength- ened to say that the only posets P0 such that for every natural num- n ber n there exists a poset Pn with id (Pn) =∼ P0 are those having ascending chain condition. On the other hand, a wide class of cases is noted where id(P ) is embeddable in P: Counterexamples are given to many variants of the statements proved. -
Forcing in the Alternative Set Theory I
Comment.Math.Univ.Carolin. 32,2 (1991)323–337 323 Forcing in the alternative set theory I Jirˇ´ı Sgall Abstract. The technique of forcing is developed for the alternative set theory (AST) and similar weak theories, where it can be used to prove some new independence results. There are also introduced some new extensions of AST. Keywords: alternative set theory, forcing, generic extension, symmetric extension, axiom of constructibility Classification: Primary 03E70; Secondary 03E25, 03E35, 03E45 We develop the method of forcing in the alternative set theory (AST) and similar weak theories like the second order arithmetic to settle some questions on indepen- dence of some axioms not included in AST. The material is divided into two parts. In this paper, the technique is developed and in its continuation (A. Sochor and J. Sgall: Forcing in the AST II, to appear in CMUC) concrete results will be proved. Most of them concern some forms of the axiom of choice not included in the basic axiomatics of AST. The main results are: (1) The axiom of constructibility is independent of AST plus the strong scheme of choice plus the scheme of dependent choices. (2) The scheme of choice is independent of A2 (the second order arithmetic). This is already known, but our proof works in A3, while the old one uses cardinals up to ℵω, which needs much stronger theory. (3) The scheme of choice is independent of AST. Let us sketch the main points different from the technique of forcing in the classical set theory: We construct generic extensions such that sets are absolute (i.e. -
Set (Mathematics) from Wikipedia, the Free Encyclopedia
Set (mathematics) From Wikipedia, the free encyclopedia A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. Contents The intersection of two sets is made up of the objects contained in 1 Definition both sets, shown in a Venn 2 Describing sets diagram. 3 Membership 3.1 Subsets 3.2 Power sets 4 Cardinality 5 Special sets 6 Basic operations 6.1 Unions 6.2 Intersections 6.3 Complements 6.4 Cartesian product 7 Applications 8 Axiomatic set theory 9 Principle of inclusion and exclusion 10 See also 11 Notes 12 References 13 External links Definition A set is a well defined collection of objects. Georg Cantor, the founder of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[1] A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] and of our thought – which are called elements of the set. The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. -
Arxiv:1204.2193V2 [Math.GM] 13 Jun 2012
Alternative Mathematics without Actual Infinity ∗ Toru Tsujishita 2012.6.12 Abstract An alternative mathematics based on qualitative plurality of finite- ness is developed to make non-standard mathematics independent of infinite set theory. The vague concept \accessibility" is used coherently within finite set theory whose separation axiom is restricted to defi- nite objective conditions. The weak equivalence relations are defined as binary relations with sorites phenomena. Continua are collection with weak equivalence relations called indistinguishability. The points of continua are the proper classes of mutually indistinguishable ele- ments and have identities with sorites paradox. Four continua formed by huge binary words are examined as a new type of continua. Ascoli- Arzela type theorem is given as an example indicating the feasibility of treating function spaces. The real numbers are defined to be points on linear continuum and have indefiniteness. Exponentiation is introduced by the Euler style and basic properties are established. Basic calculus is developed and the differentiability is captured by the behavior on a point. Main tools of Lebesgue measure theory is obtained in a similar way as Loeb measure. Differences from the current mathematics are examined, such as the indefiniteness of natural numbers, qualitative plurality of finiteness, mathematical usage of vague concepts, the continuum as a primary inexhaustible entity and the hitherto disregarded aspect of \internal measurement" in mathematics. arXiv:1204.2193v2 [math.GM] 13 Jun 2012 ∗Thanks to Ritsumeikan University for the sabbathical leave which allowed the author to concentrate on doing research on this theme. 1 2 Contents Abstract 1 Contents 2 0 Introdution 6 0.1 Nonstandard Approach as a Genuine Alternative . -
Omega-Models of Finite Set Theory
ω-MODELS OF FINITE SET THEORY ALI ENAYAT, JAMES H. SCHMERL, AND ALBERT VISSER 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 ω-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 ω-model of ZFfin that is nonstandard (i.e., not isomor- phic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In par- ticular, we present a new method for building ω-models of ZFfin that leads to a perspicuous construction of recursive nonstandard ω-models of ZFfin without the use of permutations. Furthermore, we show that every recursive model of ZFfin is an ω-model. The central theorem of the paper is the following: Theorem A. For every graph (A, F ), where F is a set of un- ordered pairs of A, there is an ω-model M of ZFfin whose universe contains A and which satisfies the following conditions: (1) (A, F ) is definable in M; (2) Every element of M is definable in (M, a)a∈A; (3) If (A, F ) is pointwise definable, then so is M; (4) Aut(M) =∼ Aut(A, F ). Theorem A enables us to build a variety of ω-models with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin. -
Some Elementary Informal Set Theory
Chapter 1 Some Elementary Informal Set Theory Set theory is due to Georg Cantor. \Elementary" in the title above does not apply to the body of his work, since he went into considerable technical depth in this, his new theory. It applies however to our coverage as we are going to restrict ourselves to elementary topics only. Cantor made many technical mistakes in the process of developing set theory, some of considerable consequence. The next section is about the easiest and most fundamental of his mistakes. How come he made mistakes? The reason is that his theory was not based on axioms and rigid rules of reasoning |a state of affairs for a theory that we loosely characterise as \informal". At the opposite end of informal we have the formal theories that are based on axioms and logic and are thus \safer" to develop (they do not lead to obvious contradictions). One cannot fault Cantor for not using logic in arguing his theorems |that process was not invented when he built his theory| but then, a fortiori, mathe- matical logic was not invented in Euclid's time either, and yet he did use axioms that stated how his building blocks, points, lines and planes interacted and be- haved! Guess what: Euclidean Geometry leads to no contradictions. The problem with Cantor's set theory is that anything goes as to what sets are and how they come about. He neglected to ask the most fundamental question: \How are sets formed?"y He just sidestepped this and simply said that a set is any collection. -
Modern Czech Logic: Vopenka and Hajek, History and Background
Modern Czech Logic: Vopěnka and Hájek, History and Background Vítězslav Švejdar∗ Charles University, Prague [email protected] Abstract We review some aspects of the history of modern Czech logic, with an emphasis on the role of Petr Vopěnka and Petr Hájek in the earlier stage of their careers. Keywords: Petr Vopěnka, Petr Hájek, Set Theory, Nonstandard Numbers, Semiset. 1 Introduction This paper is devoted to two distinguished Czech logicians, Petr Hájek and Petr Vopěnka. While Petr Hájek is well-known everywhere and has thousands of citations to his works in the WoS database, Petr Vopěnka is much less known. He did have some contacts in Poland and in Russia, but he did not travel much, and almost never left the Russia-dominated part of the world, and thus (as I was told by Petr Hájek) some of the people who knew his name even suspected that he might be a virtual person, much like Nicholas Bourbaki. Both Hájek and Vopěnka had a major influence on modern Czech (and world) logic. In this paper we will review some of their results and achievements. We will also mention their motivations and some social aspects of their work, and try to say something about their philosophical background. Vopěnka, though he had fewer Ph.D. students and fewer citations than Hájek, was nonetheless a key figure in the history of logic. We think that his early work can serve as a subject matter of an interesting historical research project. This paper could be viewed as an early step in such research. However, what is written here is based more on personal memories than on serious study of sources. -
Arxiv:2103.11387V2 [Math.LO] 4 Jul 2021 Research Ocp,Dul Ola Ler,Ctgrclduality
Double Boolean algebras with operators, Kripke contexts and some modal systems Prosenjit Howlader∗† Mohua Banerjee‡ Department of Mathematics and Statistics Indian Institute of Technology Kanpur Kanpur 208016,Uttar Pradesh, India July 6, 2021 Abstract In formal concept analysis, the collection of protoconcepts of any con- text forms a double Boolean algebra (dBa) which is fully contextual. Semi- concepts of a context form a pure dBa. The present article is a study on topological representation results for dBas, and in particular, those for fully contextual and pure dBas. The representation is in terms of object oriented protoconcepts and semiconcepts of a context. A context on topo- logical spaces (CTS) is considered, and the focus is on a special kind of CTS in which the relation defining the context as well as the converse of the relation are continuous with respect to the topologies. Such CTS are denoted as “CTSCR”. Every dBa is shown to be quasi-embeddable into the dBa of clopen object oriented protoconcepts of a particular CTSCR. The quasi-embedding turns into an embedding in case of a contextual dBa, and into an isomorphism, when the dBa is fully contextual. For pure dBas, one obtains an isomorphism with the algebra of clopen object oriented semiconcepts of the CTSCR. Representation of finite dBas and Boolean algebras is also obtained in the process. “Stone contexts” are defined next, on abstraction of properties of the CTSCR; with CTSCR- arXiv:2103.11387v2 [math.LO] 4 Jul 2021 homeomorphisms these form a category, denoted as Scxt. Pure dBas and fully contextual dBas along with dBa isomorphisms also form categories, denoted as PDBA and FCDBA respectively.