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Enumerations in computable structure theory
Enumerations in computable structure theory

... relatively computably categorical. Manasse [14] used Selivanov’s family of sets to produce a computable structure with a relation that is intrinsically computably enumerable (c.e.) but not relatively intrinsically c.e. Goncharov [10] constructed families of sets that he then used to produce computab ...
6.042J Chapter 4: Number theory
6.042J Chapter 4: Number theory

... Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . . . . Which one don’t you understand? Second, what practical value is there in it? The math ...
Chapter 1
Chapter 1

... of values to arguments. The set of all those arguments to which the function assigns values is called the domain of the function. The set of all those values that the function assigns to its arguments is called the range of the function. In the case of functions whose arguments are positive integers ...
Partition of a Set which Contains an Infinite Arithmetic (Respectively
Partition of a Set which Contains an Infinite Arithmetic (Respectively

... E-mail:[email protected] ...
Sense and denotation as algorithm and value
Sense and denotation as algorithm and value

... or even in the same language” and that “the difference between a translation and the original should properly [preserve sense]”. Some claim that faithful translation is impossible, though I have never understood exactly what this means, perhaps because I was unable to translate it faithfully into Gr ...
Complexity of Regular Functions
Complexity of Regular Functions

... papers by Alur et al. [7, 9, 8]. In those papers, the reader can find pointers to work describing the utility of regular functions in various applications in the field of computer-aided verification. Additional motivation for studying these functions comes from their connection to classical topics i ...
HONEST ELEMENTARY DEGREES AND DEGREES OF RELATIVE
HONEST ELEMENTARY DEGREES AND DEGREES OF RELATIVE

... element a of a lattice has the cupping property if it cups to every b > a. An element b of a lattice with 0 has the anti-cupping property if there is an a with 0 < a < b that does not cup to b. So, if a, b > 0 in a lattice with 0, a witnesses that b has the anti-cupping property if and only if b wit ...
SETS, RELATIONS AND FUNCTIONS
SETS, RELATIONS AND FUNCTIONS

... A set is a collection of well defined objects. For a collection to be a set it is necessary that it should be well defined. The word well defined was used by the German Mathematician George Cantor (1845- 1918 A.D) to define a set. He is known as father of set theory. Now-a-days set theory has become ...
Total recursive functions that are not primitive recursive
Total recursive functions that are not primitive recursive

MATH 337 Cardinality
MATH 337 Cardinality

... and set theory cannot disprove the hypothesis. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axioms of set theory. In other words, set theory also cannot prove the hypothesis. Therefore, the continuum hypothesis can be accepted as true or it can be accepted as false ...
PDF
PDF

... we include systems stronger than arithmetical comprehension, but these will play no part in this paper. Details, general background, and results, as well as many examples of reversals, can be found in Simpson [1999], the standard text on reverse mathematics. Each of the systems is given in the langu ...
Intermediate Logic
Intermediate Logic

... shorthands for specifying sets and expressing relationships between them. Often it will also be necessary to prove claims about such relationships. If you’re not familiar with mathematical proofs, this may be new to you. So we’ll walk through a simple example. We’ll prove that for any sets X and Y, ...
Transfinite progressions: A second look at completeness.
Transfinite progressions: A second look at completeness.

... reflection-specific part of Feferman’s completeness proof depends on using somewhat opaque non-standard definitions of the axioms of a theory. The system O. The notations in O are defined by a simultaneous induction together with the partial ordering
Mathematical Logic Fall 2004 Professor R. Moosa Contents
Mathematical Logic Fall 2004 Professor R. Moosa Contents

... Mathematical Logic is the study of the type of reasoning done by mathematicians. (i.e. proofs, as opposed to observation) Axioms are the first unprovable laws. They are statements about certain “basic concepts” (undefined first concepts). There is usually some sort of “soft” justification for believ ...
The Formulae-as-Classes Interpretation of Constructive Set Theory
The Formulae-as-Classes Interpretation of Constructive Set Theory

... class. However formally, classes do not exist, and expressions involving them must be thought of as abbreviations for expressions not involving them. Classes A, B are defined to be equal if ∀x[x ∈ A ↔ x ∈ B]. We may also consider an augmentation of the language of set theory whereby we allow atomic ...
Discrete Maths - Department of Computing | Imperial College London
Discrete Maths - Department of Computing | Imperial College London

... Let A be an arbitrary finite set. One way to list all the elements of P(A) is to start with ∅, then add the sets taking one element of A at a time, then the sets talking two elements from A at a time, and so on until the whole set A is added, and P(A) is complete. P ROPOSITION 2.10 Let A be a finite ...
Some new computable structures of high rank
Some new computable structures of high rank

author`s
author`s

... same,then the following apparently much stronger condition holds: There is a constantk suchthat essentiallyany set that can be recognizednondeterministicaily in time T can be recognizeddeterministically in time Tk. We then generalizethis result in variousways.Weconclude 88 by an analogywith Post’spr ...
The First Incompleteness Theorem
The First Incompleteness Theorem

... a now quite widely used convention: italic symbols will belong to our informal mathematics, sans serif symbols belong to formal T -wffs. Logic An L-wff (well-formed formula) is closed – i.e. is a sentence – if it has no free variables; a wff is open if it has free variables. We’ll use the slang ‘k-p ...
Chapter 4, Mathematics
Chapter 4, Mathematics

... multiplication are all algorithms. In logical theory ‘decision procedure’ is equivalent to ‘algorithm’. In cookery a reliable recipe is an algorithm for producing the soup, cake, stew or whatever it is that it tells us how to cook. A computer program, if it works, embodies some sort of algorithm. On ...
Document
Document

... • So (D D) results in something very finite – a procedure! • That procedure object has the germ or seed (D D) inside it – the potential for further recursion! ...
210ch2 - Dr. Djamel Bouchaffra
210ch2 - Dr. Djamel Bouchaffra

... • We know that if a  b then f(g(a))  f(g(b)) since the composition is injective. • Since f is a function, it cannot be the case that g(a) = g(b) since then f would have two different images for the same point. • Hence, g(a)  g(b) It follows that g must be an injection. However, f need not be an i ...
PDF
PDF

... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
If T is a consistent theory in the language of arithmetic, we say a set
If T is a consistent theory in the language of arithmetic, we say a set

... if a prime divides a product it divides one of its factors, and that if two numbers with no common prime factor both divide a number, then so does their product. (The reader may recognize these as results we took for granted in the proof of Lemma 16.5.) Once we have enough elementary lemmas, we can ...
Sets
Sets

...  If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is x P ( x ) , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at l ...
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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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