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Cardinality
Cardinality

CHAPTER 1 Sets - people.vcu.edu
CHAPTER 1 Sets - people.vcu.edu

... Given two sets A and B, it is possible to “multiply” them to produce a new set denoted as A × B. This operation is called the Cartesian product. To understand it, we must first understand the idea of an ordered pair. Definition 1.1 An ordered pair is a list ( x, y) of two things x and y, enclosed in ...
Math 320 Course Notes Chapter 7
Math 320 Course Notes Chapter 7

... 1. If A and B are both countably in…nite sets, then there exist bijections f : A ! N and g : B ! N: De…ne h : A B ! N N by h(a; b) = (f (a); g(b)) : Prove that h is a bijection. 2. Prove that the set of irrational numbers is an uncountable set. 3. Number 33 on page 224 in the text. 4. If A; B and C ...
Structure Programming CC112
Structure Programming CC112

... • In C, a function can call itself. Such types of functions are called recursive functions. • A function, f, is also said to be recursive if it calls another function, g, which in turn calls f. • Although it may sound strange for a function to call itself, it is in fact not so strange, as many mathe ...
Document
Document

... anything about the input/output (I/O) behavior of a program P just given it code! ...
The Future of Post-Human Mathematical Logic
The Future of Post-Human Mathematical Logic

Curriculum Guide (L2)
Curriculum Guide (L2)

On the existence of a connected component
On the existence of a connected component

... It is easier to build a connected component when the graph has only finitely many components. Theorem RCA0 proves that if a countable graph G has a finite set of vertices V0 such that every vertex in G is path connected to some vertex in V0 , then G can be decomposed into components. In particular, ...
Prolog arithmetic
Prolog arithmetic

... Each operator has a precedence value associated with it. Precedence values are used to decide which operator is carried out first. In Prolog, multiplication and division have higher precedence values than addition and subtraction. ...
CDM Recursive Functions Klaus Sutner Carnegie Mellon University
CDM Recursive Functions Klaus Sutner Carnegie Mellon University

... Machine models of computation (register machines, Turing machines) are easy to describe and very natural. Turing machines in particular have the great advantage that they “clearly” correspond to the abilities of an (idealized) human computor, see Turing’s 1936 paper. Alas, they don’t match up well w ...
Lecture 8: Back-and-forth - to go back my main page.
Lecture 8: Back-and-forth - to go back my main page.

... for all n ∈ N and all x ∈ M . A cut I ⊆e M is ω-coded from above if there is a coded f : N → M such that inf{f (n) : n ∈ N} = {x ∈ M : x < f (n) for all n ∈ N} = I. Fix a countable recursively saturated M |= PA, and I 4e M that is not ω-coded from above. Theorem 8.11 (Kossak–Kotlarski [3]). The foll ...
Hilbert`s Tenth Problem
Hilbert`s Tenth Problem

... Definition 2.12. A computable function is one that is able to be computed by a program or computing machine in a finite amount of time. The set of computable functions is the same as the set of recursive functions. 2.1. Unsolvability and the Halting Problem. As people began solving problems with the ...
Separating classes of groups by first–order sentences
Separating classes of groups by first–order sentences

... groups which have been considered in group theoretical investigations related to logic. This is part of a more general endeavor to understand the expressive power of first–order logic in group theory. Here are some results in this direction. Let us say a f.g. group H is quasi-axiomatizable if, whene ...
lect22
lect22

... is, data[first]. Or we could print the last element, that is, data[last]. Let’s print data[last]. After we print data[last], we still need to print the remaining elements in reverse order. ...
Solution
Solution

... (c) In total how many possible subsets can there be? Solution. There are in total (including the empty set): ...
REPRESENTATIONS OF THE REAL NUMBERS
REPRESENTATIONS OF THE REAL NUMBERS

... This implies p = ¢ 6 and p •Inf¢(p<, p>). Properties (2) and (3) immediately follow from the general fact 6 <~ 6' ~ r6, ~_ r6 and the characterizations of the final topologies of p, p< and p>. [] For a given representation 6 of a set M we can ask which informations about x = 6 ( p ) • M are finitely ...
LOGIC I 1. The Completeness Theorem 1.1. On consequences and
LOGIC I 1. The Completeness Theorem 1.1. On consequences and

... model of T . Does the converse hold? The question was posed in the 1920’s by David Hilbert (of the 23 problems fame). The answer is that remarkably, yes, it does! This result, known as the Completeness Theorem for first-order logic, was proved by Kurt Gödel in 1929. According to the Completeness Th ...
Section 2.3: Infinite sets and cardinality
Section 2.3: Infinite sets and cardinality

... Suppose that A and B are sets (finite or infinite). We say that A and B have the same cardinality (written |A| = |B|) if a bijective correspondence exists between A and B. In other words, A and B have the same cardinality if it’s possible to match each element of A to a different element of B in such a ...
On the strength of the finite intersection principle
On the strength of the finite intersection principle

... of an A with no computable maximal subfamily with the D2 intersection property. By Remark 1.4, it suffices to ensure, for every e, that hAΦe (j) : j ∈ ωi is not a maximal subfamily. Say Φe enumerates Ai at stage s if Φe,s (a) = i for some a ≤ s; say it enumerates Ai before Aj if Φe (a) = i and Φe (b ...
2 - Set Theory
2 - Set Theory

... What we know: A ⊂ B : if we ever know that x ∈ A, then we can conclude that x ∈ B. What we want: B ⊂ A : We will assume that x ∈ B and our job is to conclude that x ∈ A. What we’ll do: Since we wish to show that B ⊂ A, we will assume that x ∈ B, which is equivalent to x 6∈ B. Our job is to show that ...
1. Sets, relations and functions. 1.1. Set theory. We assume the
1. Sets, relations and functions. 1.1. Set theory. We assume the

... is the set A characterized by the property that x ∈ A if and only if x = ai for some i = 1, . . . , n. In particular, for any object a we let singleton a equal {a} and note that x is a member of singleton a if and only if x = a. We let ∅ = {x : x 6= x} and call this set the empty set because it has ...
Constructive Set Theory and Brouwerian Principles1
Constructive Set Theory and Brouwerian Principles1

... purpose of enhancing our account of Brouwerian intuitionism by contrast. The concept of algorithm or recursive function is fundamental to the Russian schools of Markov and Shanin. Contrary to Brouwer, this school takes the viewpoint that mathematical objects must be concrete, or at least have a cons ...
Semantics of a Sequential Language for Exact Real
Semantics of a Sequential Language for Exact Real

... Smyth powerdomain of a topological space of real numbers (which they refer to as the upper powerspace). Thus, they consider possibly non-deterministic computations of total real numbers, restricting their attention to those that happen to be deterministic. In the work reported here, we instead consi ...
Sets - ncert
Sets - ncert

Sets with dependent elements: Elaborating on Castoriadis` notion of
Sets with dependent elements: Elaborating on Castoriadis` notion of

... French language, the collection of one’s mental representations (or memories) etc., are in fact examples of collections whose (certain) members present a certain dependence to other members. We find this notion of ontological dependence interesting, and in this paper we shall focus on that and its i ...
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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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