MATH 327 - Winona State University
... Course Title: Foundations of Mathematics Number of Credits: 4 Catalog Description: With an emphasis on mathematical proof writing, the following topics are covered: structure of the real and complex numbers, elementary number theory, introductory group and field properties, basic topology of the rea ...
... Course Title: Foundations of Mathematics Number of Credits: 4 Catalog Description: With an emphasis on mathematical proof writing, the following topics are covered: structure of the real and complex numbers, elementary number theory, introductory group and field properties, basic topology of the rea ...
View Full Course Description - University of Nebraska–Lincoln
... which is in use today to secure information sent via the internet). As the number theory results are developed, connections to middle level curricula are emphasized and proofs are carefully selected so that those which are included in the course are particularly relevant and accessible to middle lev ...
... which is in use today to secure information sent via the internet). As the number theory results are developed, connections to middle level curricula are emphasized and proofs are carefully selected so that those which are included in the course are particularly relevant and accessible to middle lev ...
General Proof Theory - Matematički institut SANU
... essentially by Gödel in his famous incompleteness theorem, carried on further by Gerhard Gentzen with his cut elimination theorem. In 1957, at a famous conference in Ithaca, proof theory was recognized as one of the four pillars of mathematical logic (along with model theory, recursion theory and s ...
... essentially by Gödel in his famous incompleteness theorem, carried on further by Gerhard Gentzen with his cut elimination theorem. In 1957, at a famous conference in Ithaca, proof theory was recognized as one of the four pillars of mathematical logic (along with model theory, recursion theory and s ...
pdf
... Peano Arithmetic, which, as we have shown, can represent the computable functions over natural numbers. One may argue that this is the case because Peano Arithmetic has innitely many (induction) axioms and that a nite axiom system surely wouldn't lead to undecidability and undenability issues. In ...
... Peano Arithmetic, which, as we have shown, can represent the computable functions over natural numbers. One may argue that this is the case because Peano Arithmetic has innitely many (induction) axioms and that a nite axiom system surely wouldn't lead to undecidability and undenability issues. In ...
The Origin of Proof Theory and its Evolution
... First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithmetic is a first-order theory commonly formalized in ...
... First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithmetic is a first-order theory commonly formalized in ...
Relating Infinite Set Theory to Other Branches of Mathematics
... of Goldstein, the other a graph-theoretic algorithm of Kirby and Paris—whose termination proofs depend on showing that the state corresponds to a descending sequence of infinite ordinals. I had never seen this, and would not have thought it possible. Leaving infinite sets for a while, Stillwell turn ...
... of Goldstein, the other a graph-theoretic algorithm of Kirby and Paris—whose termination proofs depend on showing that the state corresponds to a descending sequence of infinite ordinals. I had never seen this, and would not have thought it possible. Leaving infinite sets for a while, Stillwell turn ...
PHIL012 Class Notes
... Truth Value & Reference in Set Theory • In Set Theory, once the reference of a name is fixed, the truth value of all sentences containing that name is fixed once and for all. • If a= {1} and b = { 1, 2 }, all statements about a and b will always • The only way the truth value of these statements co ...
... Truth Value & Reference in Set Theory • In Set Theory, once the reference of a name is fixed, the truth value of all sentences containing that name is fixed once and for all. • If a= {1} and b = { 1, 2 }, all statements about a and b will always • The only way the truth value of these statements co ...
Set theory, by Thomas Jech, Academic Press, New York, 1978, xii +
... different models), showing that a large number of famous open problems of set theory, beginning with CH, were in principle not decidable from ZFC. Set-theorists can now investigate the consequences of ZFC augmented by the various additional axioms which have been shown to be consistent with it. Thes ...
... different models), showing that a large number of famous open problems of set theory, beginning with CH, were in principle not decidable from ZFC. Set-theorists can now investigate the consequences of ZFC augmented by the various additional axioms which have been shown to be consistent with it. Thes ...
1 Introduction 2 What is Discrete Mathematics?
... This course covers the mathematical topics most directly related to computer science. Topics include: logic, basic set theory, proof techniques, number theory, mathematical induction, recursion, recurrence relations, counting, probability and graph theory. Emphasis will be placed on providing a cont ...
... This course covers the mathematical topics most directly related to computer science. Topics include: logic, basic set theory, proof techniques, number theory, mathematical induction, recursion, recurrence relations, counting, probability and graph theory. Emphasis will be placed on providing a cont ...
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.