[Write on board:
... etc. Our approximations alternate between being less than sqrt(2) and greater than sqrt(2). (To learn more about this approach to approximating irrational numbers, take a number theory course and learn about continued fractions.) ...
... etc. Our approximations alternate between being less than sqrt(2) and greater than sqrt(2). (To learn more about this approach to approximating irrational numbers, take a number theory course and learn about continued fractions.) ...
§1. Basic definitions Let IR be the set of all real numbers, while IR
... Let IR be the set of all real numbers, while IR := IR {−∞; ∞} be an extended set of real numbers, and IN := {1, 2 . . . , n, . . .} be the set of all natural numbers. Definition 1.1. Let X be a set of elements of arbitrary nature. For any n ∈ IN, by xn we denote an element of X corresponding to n. T ...
... Let IR be the set of all real numbers, while IR := IR {−∞; ∞} be an extended set of real numbers, and IN := {1, 2 . . . , n, . . .} be the set of all natural numbers. Definition 1.1. Let X be a set of elements of arbitrary nature. For any n ∈ IN, by xn we denote an element of X corresponding to n. T ...
Lecture Notes on Stability Theory
... i.e. Boolean combinations of algebraic sets. Th (M) is axiomatized as the theory of algebraically closed fields of char 0, denoted ACF0 . Note in particular that every definable subset of M is either finite, or cofinite. Theories satisfying this property are called strongly minimal. (2) Let M = (R, ...
... i.e. Boolean combinations of algebraic sets. Th (M) is axiomatized as the theory of algebraically closed fields of char 0, denoted ACF0 . Note in particular that every definable subset of M is either finite, or cofinite. Theories satisfying this property are called strongly minimal. (2) Let M = (R, ...
On the Number of False Witnesses for a Composite Number
... Thus, if n is composite, then F(n) is the set (in fact, group) of residues mod n that are false witnesses for n and F(n) is the number of such residues. If n is prime, then F(n) = n - 1 and F(n) is the entire group of reduced residues mod n. For any n, Lagrange’s theorem gives F(n) 1I$( n), where tp ...
... Thus, if n is composite, then F(n) is the set (in fact, group) of residues mod n that are false witnesses for n and F(n) is the number of such residues. If n is prime, then F(n) = n - 1 and F(n) is the entire group of reduced residues mod n. For any n, Lagrange’s theorem gives F(n) 1I$( n), where tp ...
SEQUENCES, CONTINUED Definition 3.13. A sequence {sn} of real
... Recall that S R is the set of extended real number subsequential limits of fsn g. Theorem 3.17. (a) s 2 S and s 2 S. (b) For each x > s there exists N 2 N such that sn < x for n N . For each x < s there exists N 2 N such that sn > x for n N . Moreover, s and s are the only numbers with properties (a ...
... Recall that S R is the set of extended real number subsequential limits of fsn g. Theorem 3.17. (a) s 2 S and s 2 S. (b) For each x > s there exists N 2 N such that sn < x for n N . For each x < s there exists N 2 N such that sn > x for n N . Moreover, s and s are the only numbers with properties (a ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".