Infinitesimals Abstract
... In particular, there are no points on the real line that can be assigned uniquely to the infinitesimal hyper-reals, or to the infinite hyper-reals, or to the non-constant hyper-reals. Thus, the real line is inadequate for Infinitesimal Calculus. Consequently, whenever only infinitesimals or infinite ...
... In particular, there are no points on the real line that can be assigned uniquely to the infinitesimal hyper-reals, or to the infinite hyper-reals, or to the non-constant hyper-reals. Thus, the real line is inadequate for Infinitesimal Calculus. Consequently, whenever only infinitesimals or infinite ...
Proof by Induction
... A proof by induction for the proposition “P(n) for every positive integer n” is nothing but a direct proof of the more complex proposition “(P(1) ∧ P(2) ∧ · · · ∧ P(n − 1)) → P(n) for every positive integer n”. Because it’s a direct proof, it must start by considering an arbitrary positive integer, ...
... A proof by induction for the proposition “P(n) for every positive integer n” is nothing but a direct proof of the more complex proposition “(P(1) ∧ P(2) ∧ · · · ∧ P(n − 1)) → P(n) for every positive integer n”. Because it’s a direct proof, it must start by considering an arbitrary positive integer, ...
Chapter 2 - Complex Numbers
... remedy a defect in the rational numbers, complex numbers were introduced to remedy a defect in the real numbers, i.e., there are no real solutions of equations such as x2 + 1 = 0. Let us start by defining complex numbers and then see how we can add, multiply and divide one complex number by another. ...
... remedy a defect in the rational numbers, complex numbers were introduced to remedy a defect in the real numbers, i.e., there are no real solutions of equations such as x2 + 1 = 0. Let us start by defining complex numbers and then see how we can add, multiply and divide one complex number by another. ...
41(3)
... In an earlier paper, the present author proved that if w is an a™word having length q, then M([w]) is a set of q consecutive positive integers and S(w) = q — 1. Each of these properties actually characterizes a-words (Theorem 4.4). The result used to prove this characterization is itself a character ...
... In an earlier paper, the present author proved that if w is an a™word having length q, then M([w]) is a set of q consecutive positive integers and S(w) = q — 1. Each of these properties actually characterizes a-words (Theorem 4.4). The result used to prove this characterization is itself a character ...
Dedekind cuts of Archimedean complete ordered abelian groups
... 133 – 134), Enriques ([10] pp. 37–38), and Hahn ([17] p. 603), the latter of whom was aware that for ordered abelian groups, Archimedean completeness implies, but is not ...
... 133 – 134), Enriques ([10] pp. 37–38), and Hahn ([17] p. 603), the latter of whom was aware that for ordered abelian groups, Archimedean completeness implies, but is not ...
