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Axiomatic Set Teory P.D.Welch.
Axiomatic Set Teory P.D.Welch.

... discourse) in which all the ZF axioms are seen to be true, and to carve out a special transitive subclass - the class of constructible sets, abbreviated by the letter L. This L was a proper transitive class of sets (it contains all ordinals) and it was shown by Gödel (i) That any axiom  from ZF wa ...
Discrete Mathematics - Harvard Mathematics Department
Discrete Mathematics - Harvard Mathematics Department

... Suppose that f is a function whose domain and codomain are subsets of the real numbers. Then f is called increasing if f (x) ≤ f (y ) strictly increasing if f (x) < f (y ) decreasing if f (x) ≥ f (y ) strictly decreasing if f (x) > f (y ) whenever x and y are in the domain of f and x < y . Remark No ...
Intuitionistic Type Theory
Intuitionistic Type Theory

... usually done following Tarski and assuming set theory. What we do here is meant to be closer to ordinary mathematical practice. We will avoid keeping form and meaning (content) apart. Instead we will at the same time display certain forms of judgement and inference that are used in mathematical proo ...
Intuitionistic Type Theory - The collected works of Per Martin-Löf
Intuitionistic Type Theory - The collected works of Per Martin-Löf

... usually done following Tarski and assuming set theory. What we do here is meant to be closer to ordinary mathematical practice. We will avoid keeping form and meaning (content) apart. Instead we will at the same time display certain forms of judgement and inference that are used in mathematical proo ...
Lecture 09
Lecture 09

... • Sometimes the recursive definition has an exclusion rule, which specifies that the set contains nothing other than those elements specified in the basis step and generated by applications of the rules in the recursive step. • We will always assume that the exclusion rule holds, even if it is not e ...
Fuzzy Sets - Computer Science | SIU
Fuzzy Sets - Computer Science | SIU

... “chairs”. At one end is a Chippendale. Next to it is a near-Chippendale, in fact indistinguishable from the first item. Succeeding “chairs” are less and less chair-like, until the line ends with a log. When does a chair become a log? Max Black stated that if a continuum is discrete, a number can be ...
Set Theory for Computer Science (pdf )
Set Theory for Computer Science (pdf )

... A brief history of sets A set is an unordered collection of objects, and as such a set is determined by the objects it contains. Before the 19th century it was uncommon to think of sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural numb ...
Introduction to Discrete Structures Introduction
Introduction to Discrete Structures Introduction

... than or equal to 100. – The cardinality of the empty set is ||=0 – The sets N, Z, Q, R are all infinite ...
CS 208: Automata Theory and Logic
CS 208: Automata Theory and Logic

... Q is the set of rational number R is the set of real numbers ...
Strong Logics of First and Second Order
Strong Logics of First and Second Order

... The first task is to render precise the relevant notion of “absoluteness”. This will be the main purpose of Sections 1 and 2. In Section 1 we shall investigate the many facets of the absoluteness of first-order logic. In Section 2 we shall start by investigating two traditional strong logics (ω-log ...
Chapter 6
Chapter 6

... Notice that the proof is almost identical to the algorithm description. This illustrates that when designing a recursive program, all smaller instances of a problem may be assumed to work correctly. The recursive program combines solutions to the smaller problems which had to be solved immediately. ...
Department of Mathematics, Jansons Institute of Technology
Department of Mathematics, Jansons Institute of Technology

... topological structures in fuzzy topological spaces to intuitionistic fuzzy topological spaces using intuitionistic fuzzy sets. Later many researchers have studied topics related to intuitionistic fuzzy topological spaces. On the other hand, Coker [3] introduced the concept of ‘intuitionistic sets’ i ...
Lecture notes on descriptional complexity and randomness
Lecture notes on descriptional complexity and randomness

... The present section can be read as an independent survey on the problems of randomness. It serves as some motivation for the dryer stuff to follow. If we toss a coin 100 times and it shows each time Head, we feel lifted to a land of wonders, like Rosencrantz and Guildenstern in [48]. The argument th ...
Notions of Computability at Higher Type
Notions of Computability at Higher Type

... §1. Introduction. This article is essentially a survey of fifty years of research on higher type computability. It was a great privilege to present much of this material in a series of three lectures at the Paris Logic Colloquium. In elementary recursion theory, one begins with the question: what do ...
Gödel`s correspondence on proof theory and constructive mathematics
Gödel`s correspondence on proof theory and constructive mathematics

... 1950’s, anyone reading Herbrand’s Lemma 3 should have been suspicious. Probably the error escaped public notice for so long simply because Herbrand’s paper itself, important though it was at that time, was badly written and seems not to have been widely studied. Alternative proofs of (versions of) H ...


... In 1965 [[7]], Zadeh first introduced the concept of fuzzy sets. In many real applications to handle uncertainty, fuzzy set is very much useful and in this one real value  A ( x) [0,1] is used to represent the grade of membership of a fuzzy set A defined on the universe of discorse X. After two de ...
Partiality and recursion in interactive theorem provers: An overview
Partiality and recursion in interactive theorem provers: An overview

... In the following, we introduce terminology and notational conventions that we use throughout this paper in the hope to ease its reading. 1.2.1. Terminology The term system refers to a proof assistant, including its logic and implementation. It defines the rules that specify which definitions and rea ...
Sets, Whole Numbers, and Numeration The Mayan Numeration
Sets, Whole Numbers, and Numeration The Mayan Numeration

... equal sets are always equivalent, since each element can be matched with itself, but that equivalent sets are not necessarily equal. For example, {1, 2} ⬃ {a, b}, but {1, 2} ⫽ {a, b}. The two sets A  {a, b} and B  {a, b, c} are not equivalent. However, they do satisfy the relationship defined next ...
Peano`s Arithmetic
Peano`s Arithmetic

... back to the publisher with corrections and his own suggestions for improvement. He even asked for permission to publish Genocchi’s lectures. After fixing some errors and adding his own comments to the collection, he only listed himself as an editor [2]. In 1884 Peano became a professor at the univer ...
The Art of Ordinal Analysis
The Art of Ordinal Analysis

... a rough outline of the underlying ideas will be discussed next. The most common logical calculi are Hilbert-style systems. They are specified by delineating a collection of schematic logical axioms and some inference rules. The choice of axioms and rules is more or less arbitrary, only subject to th ...
Set Theory - ScholarWorks@GVSU
Set Theory - ScholarWorks@GVSU

... then x 2 B” must be true since the hypothesis will always be false. Another way to look at this is to consider the following statement: ; 6 B means that there exists an x 2 ; such that x … B. However, this statement must be false since there does not exist an x in ;. Since this is false, we must co ...
Barwise: Infinitary Logic and Admissible Sets
Barwise: Infinitary Logic and Admissible Sets

... There are many natural examples of mathematical properties expressible in Lω1 ω . Let α be a countable ordinal. In the vocabulary L = {≤} of orderings, there is an Lω1 ω sentence whose models are just the orderings of type α, and there is an Lω1 ω formula saying, in a linear ordering, that the inter ...
Proof Pearl: Defining Functions Over Finite Sets
Proof Pearl: Defining Functions Over Finite Sets

... Can the argument g be eliminated? The application of g to the set elements cannot be replaced by a separate use of the image operator because the resulting collection of values must not be regarded as a set. We could combine f and g into one function, namely λx y. f (g x ) y. That approach, followe ...
Recursive Predicates And Quantifiers
Recursive Predicates And Quantifiers

... the relationship of the results stands out more clearly than before. The general theorem asserts that to each of an enumeration of predicate forms, there is a predicate not expressible in that form. The predicates considered belong to elementary number theory. The possibility that this theorem may a ...
Finite and Infinite Sets
Finite and Infinite Sets

... We will use the concept of equivalent sets introduced in Preview Activity 1 to define a finite set. Definition. A set A is a finite set provided that A D ; or there exists a natural number k such that A  Nk . A set is an infinite set provided that it is not a finite set. If A  Nk , we say that the ...
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Computability theory

Computability theory, also called recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.The basic questions addressed by recursion theory are ""What does it mean for a function on the natural numbers to be computable?"" and ""How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?"". The answers to these questions have led to a rich theory that is still being actively researched. The field has since grown to include the study of generalized computability and definability. Invention of the central combinatorial object of recursion theory, namely the Universal Turing Machine, predates and predetermines the invention of modern computers. Historically, the study of algorithmically undecidable sets and functions was motivated by various problems in mathematics that turned to be undecidable; for example, word problem for groups and the like. There are several applications of the theory to other branches of mathematics that do not necessarily concentrate on undecidability. The early applications include the celebrated Higman's embedding theorem that provides a link between recursion theory and group theory, results of Michael O. Rabin and Anatoly Maltsev on algorithmic presentations of algebras, and the negative solution to Hilbert's Tenth Problem. The more recent applications include algorithmic randomness, results of Soare et al. who applied recursion-theoretic methods to solve a problem in algebraic geometry, and the very recent work of Slaman et al. on normal numbers that solves a problem in analytic number theory.Recursion theory overlaps with proof theory, effective descriptive set theory, model theory, and abstract algebra.Arguably, computational complexity theory is a child of recursion theory; both theories share the same technical tool, namely the Turing Machine. Recursion theorists in mathematical logic often study the theory of relative computability, reducibility notions and degree structures described in this article. This contrasts with the theory of subrecursive hierarchies, formal methods and formal languages that is common in the study of computability theory in computer science. There is a considerable overlap in knowledge and methods between these two research communities; however, no firm line can be drawn between them. For instance, parametrized complexity was invented by a complexity theorist Michael Fellows and a recursion theorist Rod Downey.
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