Chapter 0. Introduction to the Mathematical Method
... A sufficient condition for a natural number to be a multiple of 360 is that it is a multiple of 3 and 120. Proof Let there be the following propositions: A: To be multiple of 360 B1: To be multiple of 3 B2: To be multiple of 120 B: B1 ∩ B2 We need to prove that B ⇒ A (B is sufficient for A). Proof B ...
... A sufficient condition for a natural number to be a multiple of 360 is that it is a multiple of 3 and 120. Proof Let there be the following propositions: A: To be multiple of 360 B1: To be multiple of 3 B2: To be multiple of 120 B: B1 ∩ B2 We need to prove that B ⇒ A (B is sufficient for A). Proof B ...
Scattered Sentences have Few Separable Randomizations
... This note answers a question posed in the paper [K2], and results from a discussion following a lecture by Keisler at the Midwest Model Theory meeting in Chicago on April 5, 2016. Fix a countable first order signature L. A sentence ϕ of the infinitary logic Lω1 ω is scattered if there is no countabl ...
... This note answers a question posed in the paper [K2], and results from a discussion following a lecture by Keisler at the Midwest Model Theory meeting in Chicago on April 5, 2016. Fix a countable first order signature L. A sentence ϕ of the infinitary logic Lω1 ω is scattered if there is no countabl ...
First-Order Proof Theory of Arithmetic
... This chapter discusses the proof-theoretic foundations of the first-order theory of the non-negative integers. This first-order theory of numbers, also called ‘first-order arithmetic’, consists of the first-order sentences which are true about the integers. The study of first-order arithmetic is imp ...
... This chapter discusses the proof-theoretic foundations of the first-order theory of the non-negative integers. This first-order theory of numbers, also called ‘first-order arithmetic’, consists of the first-order sentences which are true about the integers. The study of first-order arithmetic is imp ...
Gödel Without (Too Many) Tears
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
The History of Categorical Logic
... Categorical logic, as its name indicates, is logic in the setting of category theory. But this description does not say much. Most readers would probably find more instructive to learn that categorical logic is algebraic logic, pure and simple. It is logic in an algebraic dressing. Just as algebraic ...
... Categorical logic, as its name indicates, is logic in the setting of category theory. But this description does not say much. Most readers would probably find more instructive to learn that categorical logic is algebraic logic, pure and simple. It is logic in an algebraic dressing. Just as algebraic ...
Set Theory for Computer Science (pdf )
... • Mathematical argument: Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. • Sets and logic: Subsets of a fixed set as a Boolean algebra. Venn diagrams. Propositional logic and its models. Validity, entailment, and equivalence of boo ...
... • Mathematical argument: Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. • Sets and logic: Subsets of a fixed set as a Boolean algebra. Venn diagrams. Propositional logic and its models. Validity, entailment, and equivalence of boo ...
Nonmonotonic Reasoning - Computer Science Department
... by usual type reasoning systems, except that the rules carry the list of “exceptional cases” making the application of such rule invalid. Formally, Reiter, [25] introduced the concept of default theory. A default theory is a pair hD, W i where W is a set of sentences of the underlying language L and ...
... by usual type reasoning systems, except that the rules carry the list of “exceptional cases” making the application of such rule invalid. Formally, Reiter, [25] introduced the concept of default theory. A default theory is a pair hD, W i where W is a set of sentences of the underlying language L and ...
Proof Pearl: Defining Functions over Finite Sets
... We assume there already is a formalization of sets, with standard operations such as comprehension, union and intersection. In higher-order logic, this is trivial by the obvious representation of sets by predicates. We use standard mathematical notation with a few extensions. Type variables are writ ...
... We assume there already is a formalization of sets, with standard operations such as comprehension, union and intersection. In higher-order logic, this is trivial by the obvious representation of sets by predicates. We use standard mathematical notation with a few extensions. Type variables are writ ...
Set theory and logic
... Prejudices against this viewpoint were responsible for the rejection of his work by some mathematicians, but others reacted favorably because the theory provided a proof of the existence of transcendental numbers. Other applications in analysis and geometry were found, and Cantor's theory of sets wo ...
... Prejudices against this viewpoint were responsible for the rejection of his work by some mathematicians, but others reacted favorably because the theory provided a proof of the existence of transcendental numbers. Other applications in analysis and geometry were found, and Cantor's theory of sets wo ...
Department of Mathematics, Jansons Institute of Technology
... of fuzzy sets, it became a popular topic of investigation in the fuzzy set community. Many mathematical advantages of intuitionistic fuzzy sets have been discussed. Coker [2] generalised topological structures in fuzzy topological spaces to intuitionistic fuzzy topological spaces using intuitionisti ...
... of fuzzy sets, it became a popular topic of investigation in the fuzzy set community. Many mathematical advantages of intuitionistic fuzzy sets have been discussed. Coker [2] generalised topological structures in fuzzy topological spaces to intuitionistic fuzzy topological spaces using intuitionisti ...
Slide 1
... Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs The extension of truth operations for tautologies, contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for ...
... Fuzzy Tautologies, Contradictions, Equivalence, and Logical Proofs The extension of truth operations for tautologies, contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for ...
Introduction to Discrete Structures Introduction
... Set equivalences (cheat sheet or Table 1, page 124) ...
... Set equivalences (cheat sheet or Table 1, page 124) ...
Partial Grounded Fixpoints
... generalise them to points in the bilattice, while still maintaining the elegance and desirable properties of groundedness. For the case of logic programming, this generalisation boils down to extending groundedness to partial (or three-valued) interpretations. There are several reasons why it is imp ...
... generalise them to points in the bilattice, while still maintaining the elegance and desirable properties of groundedness. For the case of logic programming, this generalisation boils down to extending groundedness to partial (or three-valued) interpretations. There are several reasons why it is imp ...
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.