Notes from [Kevin O'Neill: An Introduction to Non-Standard Analysis and its Applications]

[elim]

1 Basic Tools

A Shortest Possible History of Analysis

When Newton and Leibnitz practiced calculus, they used innitesimals, which were supposed to be like real numbers, yet of smaller magnitude than any other type of postive real number. In the 19th century, mathematicians realized they could not justify the use of innitesimals according to their sense of rigor, so they began using denitions involving $\epsilon$'s and $\delta$'s. Then, in the early 1960's, the logician Abraham Robinson gured out a way to rigorously dene innitesimals, creating a subject now known as nonstandard analysis.

The Hyperreals

$\tau:\mathbb{R}\to \mathbb{R}^{\mathbb{N}}\;(r\mapsto (r,r,r,\ldots))$ is an injection.

$\mathbb{R}^{\mathbb{N}}$ is a ring with respect to $(a_n) + (b_n) = (a_n+b_n),\; (a_n)(b_n) = (a_n b_n)$ but not a field.

$(\frac{1+(-1)^n}{2})(\frac{1-(-1)^n}{2}) = (0,0,0,\ldots) = \tau(0)$

1.1 Definition $\mathcal{F}\subset \mathcal{P}(I)$ (set $I\ne \varnothing$) is a filter if
$\qquad$ 1. $(A,\,B\in\mathcal{F})\implies (A\cap B\in \mathcal{F})$
$\qquad$ 2. $(A \in\mathcal{F})\wedge (A\subset B\subset I)\implies (B\in \mathcal{F})$
$\qquad\;\,\mathcal{F}$ is an ultrafilter if it's a filter and $\forall A\subset I\;(A\not\in\mathcal{F})\implies (A^c \in\mathcal{F})$
$\qquad$ An ultrafilter $\mathcal{F}$ is principal if $\exists i\in I\;(\mathcal{F} = \{A\subset I: i\in A\})$