lect13 - Kent State University
... • The proof of the undecidability of the halting problem uses a technique called diagonalization, discovered first by mathematician Georg Cantor in 1873. • Cantor was concerned with the problem of measuring the sizes of infinite sets. If we have two infinite sets, how can we tell whether one is larg ...
... • The proof of the undecidability of the halting problem uses a technique called diagonalization, discovered first by mathematician Georg Cantor in 1873. • Cantor was concerned with the problem of measuring the sizes of infinite sets. If we have two infinite sets, how can we tell whether one is larg ...
The Development of Categorical Logic
... taking as an axiom the existence of an object of truth values: the result was a concept of amazing fertility, that of elementary topos—a cartesian closed category equipped with an object of truth values1. In addition to providing a natural generalization of elementary— i.e., first-order—theories, th ...
... taking as an axiom the existence of an object of truth values: the result was a concept of amazing fertility, that of elementary topos—a cartesian closed category equipped with an object of truth values1. In addition to providing a natural generalization of elementary— i.e., first-order—theories, th ...
The Surprise Examination Paradox and the Second Incompleteness
... (depending on the theory and on the programming language that is used to define Kolmogorov complexity) such that, for any integer x, the statement “K(x) > L” cannot be proved within the theory. The proof given by Chaitin is as follows. Let L be a large enough integer. Assume for a contradiction that ...
... (depending on the theory and on the programming language that is used to define Kolmogorov complexity) such that, for any integer x, the statement “K(x) > L” cannot be proved within the theory. The proof given by Chaitin is as follows. Let L be a large enough integer. Assume for a contradiction that ...
EMBEDDING AN ANALYTIC EQUIVALENCE RELATION IN THE
... In applications to Bayesian decision theory and game theory, it is reasonable to specify each agent’s information as a ∆11 (that is, Borel) equivalence relation, or even as a smooth or closed Borel relation.5 Thus it may be asked: if the graphs of E1 and E2 are in ∆11 or in some smaller class, then ...
... In applications to Bayesian decision theory and game theory, it is reasonable to specify each agent’s information as a ∆11 (that is, Borel) equivalence relation, or even as a smooth or closed Borel relation.5 Thus it may be asked: if the graphs of E1 and E2 are in ∆11 or in some smaller class, then ...
Tactical and Strategic Challenges to Logic (KAIST
... to be fruitfully applicable to inconsistent systems that might not be as big as Five Eyes, banking or health-care. Most information-systems that aren’t at all small aren’t big in the Five Eyes sense. All the same, they can be a lot bigger than we might think. The IR project is founded on assumptions ...
... to be fruitfully applicable to inconsistent systems that might not be as big as Five Eyes, banking or health-care. Most information-systems that aren’t at all small aren’t big in the Five Eyes sense. All the same, they can be a lot bigger than we might think. The IR project is founded on assumptions ...
chapter 1 set theory - New Age International
... The main object of this book is to introduce the basic algebraic systems (mathematical systems)—groups, ring, integral domains, fields, and vector spaces. By an algebraic system we shall mean a non-empty set together with one or more than one binary operations defined on the set. The systems subsequ ...
... The main object of this book is to introduce the basic algebraic systems (mathematical systems)—groups, ring, integral domains, fields, and vector spaces. By an algebraic system we shall mean a non-empty set together with one or more than one binary operations defined on the set. The systems subsequ ...
The Number of Topologies on a Finite Set
... Sharp [1] proved this result using graph theory. Stephen’s proof [4]used topological facts. Here is another one, which is direct and allows us to compute T (n, 3 · 2n−2 ). Proof. Since we are looking for a non-discrete topology τ having the maximum of open sets, it must not contain all the singleton ...
... Sharp [1] proved this result using graph theory. Stephen’s proof [4]used topological facts. Here is another one, which is direct and allows us to compute T (n, 3 · 2n−2 ). Proof. Since we are looking for a non-discrete topology τ having the maximum of open sets, it must not contain all the singleton ...
pdf format
... A good example of a class is the class V of all sets, defined by V = {x : x = x}. V is called the universe. The class ON is defined by ON = {x : “x is an ordinal”}. The term “collection” in the previous definition refers to some intuitive notion of collection, or gathering together. Note that some c ...
... A good example of a class is the class V of all sets, defined by V = {x : x = x}. V is called the universe. The class ON is defined by ON = {x : “x is an ordinal”}. The term “collection” in the previous definition refers to some intuitive notion of collection, or gathering together. Note that some c ...
PowerPoint file for CSL 02, Edinburgh, UK
... proved for the n-level. The conjecture have been solved for n=1, 2 levels, which include all of the LCM semiclassical principles. It is still open for the higher levels. ...
... proved for the n-level. The conjecture have been solved for n=1, 2 levels, which include all of the LCM semiclassical principles. It is still open for the higher levels. ...
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Logic, Sets, and Proofs
... x from the fixed set U , then there are two basic types of quantifiers: • ∀x ∈ U (P (x)). This universal quantifier means that for all (or for every or for each or for any) value of x in U , P (x) is true. Example: ∀x ∈ R (2x = (x + 1) + (x − 1)). • ∃x ∈ U (P (x)). This existential quantifier means ...
... x from the fixed set U , then there are two basic types of quantifiers: • ∀x ∈ U (P (x)). This universal quantifier means that for all (or for every or for each or for any) value of x in U , P (x) is true. Example: ∀x ∈ R (2x = (x + 1) + (x − 1)). • ∃x ∈ U (P (x)). This existential quantifier means ...
PRESENTATION OF NATURAL DEDUCTION R. P. NEDERPELT
... designed by N.G. de Bruijn for rendering mathematical texts in a formal way (see [1]). Various versions of this language have been developed by de Bruijn, in cooperation with, among others, D.T. van Daalen, L.S. Jutting and J. Zucker (see [2]). Most of the features of these various versions will be ...
... designed by N.G. de Bruijn for rendering mathematical texts in a formal way (see [1]). Various versions of this language have been developed by de Bruijn, in cooperation with, among others, D.T. van Daalen, L.S. Jutting and J. Zucker (see [2]). Most of the features of these various versions will be ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.