lecture notes on mathematical induction
... is a fundamental part of the structure of the positive integers that every positive integer can be reached in this way, i.e., starting from 1 and adding 1 sufficiently many times. In other words, any rigorous definition of the natural numbers (for instance in terms of sets, as alluded to earlier in ...
... is a fundamental part of the structure of the positive integers that every positive integer can be reached in this way, i.e., starting from 1 and adding 1 sufficiently many times. In other words, any rigorous definition of the natural numbers (for instance in terms of sets, as alluded to earlier in ...
LOGIC I 1. The Completeness Theorem 1.1. On consequences and
... x1 < x2 . Then N is an L-structure with the natural interpretation for <. Let i(xi ) = i − 1. Then hN, ii is a model of T . On the other hand, if we define j such that j(x1 ) = 2 and j(x2 ) = 0, then the pair hN, ji is not a model of T . Next, we must decide on what are the axioms of logic. The naiv ...
... x1 < x2 . Then N is an L-structure with the natural interpretation for <. Let i(xi ) = i − 1. Then hN, ii is a model of T . On the other hand, if we define j such that j(x1 ) = 2 and j(x2 ) = 0, then the pair hN, ji is not a model of T . Next, we must decide on what are the axioms of logic. The naiv ...
Fibonacci notes
... We let N denote the natural numbers, starting at zero, and N+ = N \ {0} the positive integers. Also, P is the set of all nonempty finite subsets of N ordered lexicographically. We shall often identify elements of P with finite binary words: every set X ∈ P is identified with the word εm . . . ε0 whe ...
... We let N denote the natural numbers, starting at zero, and N+ = N \ {0} the positive integers. Also, P is the set of all nonempty finite subsets of N ordered lexicographically. We shall often identify elements of P with finite binary words: every set X ∈ P is identified with the word εm . . . ε0 whe ...
An Introduction to Mathematical Logic
... 2. the claim that ϕ follows logically from Φ 3. the proof of ϕ on the basis of Φ In mathematical logic this is made precise: 1. sentences: members of so-called first-order languages 2. consequence: a first-order sentences ϕ follows logically from a set Φ of first-order sentences iff every model that ...
... 2. the claim that ϕ follows logically from Φ 3. the proof of ϕ on the basis of Φ In mathematical logic this is made precise: 1. sentences: members of so-called first-order languages 2. consequence: a first-order sentences ϕ follows logically from a set Φ of first-order sentences iff every model that ...
41(4)
... drawn in India ink on separate sheets of bond paper or vellum, approximately twice the size they are to appear in print. Since the Fibonacci Association has adopted ¥{ = F2 = 1, ¥n+i= Fn+Fn-i, n>2 and LL=1, L2 =3, L/z+/ = L/z+Ln-i, n>2 as the standard definitions for The Fibonacci and Lucas sequence ...
... drawn in India ink on separate sheets of bond paper or vellum, approximately twice the size they are to appear in print. Since the Fibonacci Association has adopted ¥{ = F2 = 1, ¥n+i= Fn+Fn-i, n>2 and LL=1, L2 =3, L/z+/ = L/z+Ln-i, n>2 as the standard definitions for The Fibonacci and Lucas sequence ...
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"".