On Interpretations of Arithmetic and Set Theory∗
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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. -
COMPSCI 501: Formal Language Theory Insights on Computability Turing Machines Are a Model of Computation Two (No Longer) Surpris
Insights on Computability Turing machines are a model of computation COMPSCI 501: Formal Language Theory Lecture 11: Turing Machines Two (no longer) surprising facts: Marius Minea Although simple, can describe everything [email protected] a (real) computer can do. University of Massachusetts Amherst Although computers are powerful, not everything is computable! Plus: “play” / program with Turing machines! 13 February 2019 Why should we formally define computation? Must indeed an algorithm exist? Back to 1900: David Hilbert’s 23 open problems Increasingly a realization that sometimes this may not be the case. Tenth problem: “Occasionally it happens that we seek the solution under insufficient Given a Diophantine equation with any number of un- hypotheses or in an incorrect sense, and for this reason do not succeed. known quantities and with rational integral numerical The problem then arises: to show the impossibility of the solution under coefficients: To devise a process according to which the given hypotheses or in the sense contemplated.” it can be determined in a finite number of operations Hilbert, 1900 whether the equation is solvable in rational integers. This asks, in effect, for an algorithm. Hilbert’s Entscheidungsproblem (1928): Is there an algorithm that And “to devise” suggests there should be one. decides whether a statement in first-order logic is valid? Church and Turing A Turing machine, informally Church and Turing both showed in 1936 that a solution to the Entscheidungsproblem is impossible for the theory of arithmetic. control To make and prove such a statement, one needs to define computability. In a recent paper Alonzo Church has introduced an idea of “effective calculability”, read/write head which is equivalent to my “computability”, but is very differently defined. -
Axiomatic Set Teory P.D.Welch
Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L. -
Algorithms, Turing Machines and Algorithmic Undecidability
U.U.D.M. Project Report 2021:7 Algorithms, Turing machines and algorithmic undecidability Agnes Davidsdottir Examensarbete i matematik, 15 hp Handledare: Vera Koponen Examinator: Martin Herschend April 2021 Department of Mathematics Uppsala University Contents 1 Introduction 1 1.1 Algorithms . .1 1.2 Formalisation of the concept of algorithms . .1 2 Turing machines 3 2.1 Coding of machines . .4 2.2 Unbounded and bounded machines . .6 2.3 Binary sequences representing real numbers . .6 2.4 Examples of Turing machines . .7 3 Undecidability 9 i 1 Introduction This paper is about Alan Turing's paper On Computable Numbers, with an Application to the Entscheidungsproblem, which was published in 1936. In his paper, he introduced what later has been called Turing machines as well as a few examples of undecidable problems. A few of these will be brought up here along with Turing's arguments in the proofs but using a more modern terminology. To begin with, there will be some background on the history of why this breakthrough happened at that given time. 1.1 Algorithms The concept of an algorithm has always existed within the world of mathematics. It refers to a process meant to solve a problem in a certain number of steps. It is often repetitive, with only a few rules to follow. In more recent years, the term also has been used to refer to the rules a computer follows to operate in a certain way. Thereby, an algorithm can be used in a plethora of circumstances. The word might describe anything from the process of solving a Rubik's cube to how search engines like Google work [4]. -
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 . -
Undecidable Problems: a Sampler (.Pdf)
UNDECIDABLE PROBLEMS: A SAMPLER BJORN POONEN Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathemat- ics. 1. Introduction The goal of this survey article is to demonstrate that undecidable decision problems arise naturally in many branches of mathematics. The criterion for selection of a problem in this survey is simply that the author finds it entertaining! We do not pretend that our list of undecidable problems is complete in any sense. And some of the problems we consider turn out to be decidable or to have unknown decidability status. For another survey of undecidable problems, see [Dav77]. 2. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number @0 strictly between @0 and 2 , is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [G¨od40,Coh63, Coh64].) The first examples of statements independent of a \natural" axiom system were constructed by K. G¨odel[G¨od31]. 2. Decision problem: A family of problems with YES/NO answers is called unde- cidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert's tenth problem, to decide whether a mul- tivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 2.1. -
Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected]
Boise State University ScholarWorks Mathematics Undergraduate Theses Department of Mathematics 5-2014 Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Boise State University, [email protected] Follow this and additional works at: http://scholarworks.boisestate.edu/ math_undergraduate_theses Part of the Set Theory Commons Recommended Citation Rahim, Farighon Abdul, "Axioms of Set Theory and Equivalents of Axiom of Choice" (2014). Mathematics Undergraduate Theses. Paper 1. Axioms of Set Theory and Equivalents of Axiom of Choice Farighon Abdul Rahim Advisor: Samuel Coskey Boise State University May 2014 1 Introduction Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set. In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite. -
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 . -
Compact Covering and Game Determinacy
&g-l.& __ TOPOLOGY I-- AND ITS EJ APPLKATIONS ELSEWIER Topology and its Applications 68 (1996) 153-185 Compact covering and game determinacy Gabriel Debs *, Jean Saint Raymond Equip d’Anulye, UnivemirP Puris 6. Boite 186, 4 pluce Jussieu, F-75252, Prtris Cedex 05, France Received 2 December 1994; revised 13 February 1995, 7 June 1995 Abstract All spaces are separable and metrizable. Suppose that the continuous and onto mapping f : X 4 Y is compact covering. Under the axiom of xi-determinacy, we prove that f is inductively perfect whenever X is Bore], and it follows then that Y is also Bore]. Under the axiom Nf = NI we construct examples showing that the conclusion might fail if “X is Borel” is replaced by “X is coanalytic”. If we suppose that both X and Y are Bore], then we prove (in ZFC) the weaker conclusion that f has a Bore1 (in fact a Baire-1) section g : Y + X. We also prove (in ZFC) that if we suppose only X to be Bore1 but of some “low” class, then Y is also Bore1 of the same class. Other related problems are discussed. Keyworcls: Compact covering; Inductively perfect; Determinacy AMS classification: 03E60; 04A15; 54E40; 54H05 Everywhere in this paper f : X -+ Y denotes a continuous and onto mapping between the spaces X and Y. By space we always mean a separable and metrizable space. We are interested in the comparison of the following two properties for f: (CC): “V L compact c Y, 3 K compact c X such that f(K) = L” and (IP): “3 X’ c X such that f(X’) = Y and V L compact c Y, the set X’ n f -’ (L) is compact”. -
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.