Topic
Ordinal arithmetic
About: Ordinal arithmetic is a research topic. Over the lifetime, 258 publications have been published within this topic receiving 3567 citations.
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Papers
Book•
30 Oct 2012
TL;DR: A list of axioms and operations for the algebra of sets can be found in this paper, where the authors also provide a discussion of the relation between the Axiom of Choice Equinumerosity Finite Sets and cardinal numbers.
Abstract: Contents Preface List of Symbols Chapter 1 Introduction Baby Set Theory Sets-An Informal View Classes Axiomatic Method Notation Historical Notes Chapter 2 Axioms and Operations Axioms Arbitrary Unions and Intersections Algebra of Sets Epilogue Review Exercises Chapter 3 Relations and Functions Ordered Pairs Relations n-Ary Relations Functions Infinite Cartesian Products Equivalence Relations Ordering Relations Review Exercises Chapter 4 Natural Numbers Inductive Sets Peano's Postulates Recursion on Arithmetic Ordering on Review Exercises Chapter 5 Construction of the Real Numbers Integers Rational Numbers Real Numbers Summaries Two Chapter 6 Cardinal Numbers and the Axiom of Choice Equinumerosity Finite Sets Cardinal Arithmetic Ordering Cardinal Numbers Axiom of Choice Countable Sets Arithmetic of Infinite Cardinals Continuum Hypothesis Chapter 7 Orderings and Ordinals Partial Orderings Well Orderings Replacement Axioms Epsilon-Images Isomorphisms Ordinal Numbers Debts Paid Rank Chapter 8 Ordinals and Order Types Transfinite Recursion Again Alephs Ordinal Operations Isomorphism Types Arithmetic of Order Types Ordinal Arithmetic Chapter 9 Special Topics Well-Founded Relations Natural Models Cofinality Appendix Notation, Logic, and Proofs Selected References for Further Study List of Axioms Index
327 citations
Book•
1 Jan 2007
TL;DR: The Square-bracket operation on countable ordinals has been studied extensively in the literature, e.g. in this article, where the authors describe a general walk on a countable cardinal.
Abstract: Walks on Countable Ordinals.- Metric Theory of Countable Ordinals.- Coherent Mappings and Trees.- The Square-bracket Operation on Countable Ordinals.- General Walks and Their Characteristics.- Square Sequences.- The Oscillation Mapping and the Square-bracket Operation.- Unbounded Functions.- Higher Dimensions.
180 citations
1 Jan 1990
TL;DR: In this paper, the authors introduce the following operations: On X that yields the set of all ordinals which belong to the set X, Lim X and sup X that yield the minimal ordinals belonging to X and the minimal min ordinals greater than all min-ordinals in X, respectively.
Abstract: Summary. In the first part of the article we introduce the following operations: On X that yields the set of all ordinals which belong to the set X, Lim X that yields the set of all limit ordinals which belong to X, and inf X and sup X that yield the minimal ordinal belonging to X and the minimal ordinal greater than all ordinals belonging to X, respectively. The second part of the article starts with schemes that can be used to justify the correctness of definitions based on the transfinite induction (see [1] or [3]). The schemes are used to define addition, product and power of ordinal numbers. The operations of limes inferior and limes superior of sequences of ordinals are defined and the concepts of limet of ordinal sequence and increasing and continuous sequence are introduced.
147 citations
TL;DR: This contribution focuses on ordinal sums of t-norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets.
137 citations
Book•
13 Oct 2008
TL;DR: In this paper, a notation system for the Ordinals below epsilon nought is presented, which is based on the Kripke Platek Set Theory with infinity.
Abstract: 1 Historical Background.- 2 Primitive Recursive Functions and Relations.- 2.1 Primitive Recursive Functions.- 2.2 Primitive Recursive Relations.- 3 Ordinals.- 3.1 Heuristic.- 3.2 Some Basic Facts on Ordinals.- 3.3 Fundamentals of Ordinal Arithmetic.- 3.3.1 A Notation System for the Ordinals below epsilon nought.- 3.4 The Veblen Hierarchy.- 3.4.1 Preliminaries.- 3.4.2 The Veblen Hierarchy.- 3.4.3 A Notation System for the Ordinals below Gamma nought.- 4 Pure Logic.- 4.1 Heiristics.- 4.2 First and Second Order Logic.- 4.3 The Tait calculus.- 4.4 Trees and the Completeness Theorem.- 4.5 Gentzens Hauptsatz for Pure First Order Logic.- 4.6 Second Order Logic.- 5 Truth Complexities for Pi 1-1-Sentences.- 5.1 The language of Arithmetic.- 5.2 The Tait language for Second Order Arithmetic.- 5.3 Truth Complexities for Arithmetical Sentences.- 5.4 Truth Complexities for Pi 1-1-Sentences.- 6 Inductive Definitions.- 6.1 Motivation.- 6.2 Inductive Definitions as Monotone Operators.- 6.3 The Stages of an Inductive Definition.- 6.4 Arithmetically Definable Inductive Definitions.- 6.5 Inductive Definitions, Well-Orderings and Well-Founded Trees.- 6.6 Inductive Definitions and Truth Complexities.- 6.7 The Pi-1-1- Ordinal of a Theory.- 7 The Ordinal Analysis for Pean Arithmetic.- 7.1 The Theory PA.- 7.2 The Theory NT.- 7.3 The Upper Bound.- 7.4 The Lower Bound.- 7.5 The Use of Gentzen's Consistency Proof for Hilbert's Programme.- 7.5.1 On the Consistency of Formal and Semi-Formal Systems.- 7.5.2 The Consistency of NT.- 7.5.3 Kreisel's Counterexample.- 7.5.4 Gentzen's Consistency Proof in the Light of Hilbert's Programme.- 8 Autonomous Ordinals and the Limits of Predicativity.- 8.1 The Language L-kappa.- 8.2 Semantics for L-kappa.- 8.3 Autonomous Ordinals.- 8.4 The Upper Bound for Autonomous Ordinals.- 8.5 The Lower Bound for Autonomous Ordinals.- 9 Ordinal Analysis of the Theory for Inductive Definitions.- 9.1 The Theory ID1.- 9.2 The Language L infinity (NT).- 9.3 The Semi-FormalSystem for L infinity (NT).- 9.3.1 Semantical Cut-Elimination.- 9.3.2 Operator Controlled Derivations.- 9.4 The Collapsing Theorem for ID1.- 9.5 The Upper Bound.- 9.6 The Lower Bound.- 9.6.1 Coding Ordinals in L(NT).- 9.6.2 The Well-Ordering Proof.- 9.7 Alternative Interpretations for Omega.- 10 Provably Recursive Functions of NT.- 10.1 Provably Recursive Functions of a Theory.- 10.2 Operator Controlled Derivations.- 10.3 Iterating Operators.- 10.4 Cut Elimination for Operator Controlled Derivations.- 10.5 The Embedding of NT.- 10.6 Discussion.- 11 Ordinal Analysis for Kripke Platek Set Theory with infinity.- 11.1 Naive Set Theory.- 11.2 The Language of Set Theory.- 11.3 Constructible Sets.- 11.4 Kripke Platek Set Theory.- 11.5 ID1 as a Subtheory of Kp-omega.- 11.6 Variations of KP-omega and Axiom beta.- 11.7 The Sigma Ordinal of KP-omega.- 11.8 The Theory of Pi-2 Reflection.- 11.9 An Infinite Verification Calculus for the Constructible Hierarchy.- 11.10 A Semi-Formal System for Ramified Set Theory.- 11.11 The Collapsing Theorem for Ramified Set Theory.- 11.12 Ordinal Analysis for Kripke Platek Set Theory.- 12 Predicativity Revisited.- 12.1 Admissible Extensions.- 12.2 M-Logic.- 12.3 Extending Semi-Formal Systems.- 12.4 Asymmetric Interpretations.- 12.5 Reduction of T+ to T.- 12.6 The Theories KP n and KP 0-n.- 12.7 The Theories KPl 0 and KP i 0.- 13 Non-Monotone Inductive Definitions.- 13.1 Non-Monotone Inductive Definitions.- 13.2 Prewellorderings.- 13.3 The Theory for Pi 0-1 definable Fixed-Points.- 13.4 ID1 as a Sub-Theory of the Theory for Pi 0-1 definable Fixed-Points.- 13.5 The Upper Bound for the Proof theoretical Ordinal of Pi 0-1-FXP.- 14 Epilogue.
119 citations
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Performance Metrics
| Year | Papers |
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| 2021 | 1 |
| 2020 | 2 |
| 2019 | 3 |
| 2018 | 1 |
| 2017 | 6 |
| 2016 | 5 |