Hyperreal numbers

Construction of the hyperreals

There's different ways to construct the hyperreals, but the simplest one (that doesn't involve extending the ZFC axiom set) is done by using sequences of real numbers.

\begin{equation} ^* \mathbb{R} = \mathbb{R}^{\mathbb{N}} / \sim \end{equation}

A hyperreal number is an equivalence class of real sequences. The equivalence relation involves ultrafilters

An ultrafilter $F$ is a subset of a partially ordered set $(P, \leq)$ such that

$F \neq \varnothing$

Ultrafilters aren't actually constructible, but for the most part this won't be necessary. The main issue will be for sequences of the type

\begin{equation} (0,1,0,1,0,1,...) \end{equation}

which have an infinite number of more than one value, and no clear pattern. Depending on the ultrafilter used, this sequence will have the value of either $0$ or $1$.

Our equivalence relation is the following : two sequences $a,b$ are equivalent (noted $a \sim_F b$) if

\begin{equation} \{ i \in \mathbb{N} | a_i = b_i \} \in F \end{equation}

Therefore, as a set, the hyperreals are simply $\mathbb{R}^{\mathbb{N}} / \sim_F$

Operations on hyperreal numbers are performed term by term in the sequences, that is,

\begin{equation} a + b = (a_1 + b_1, a_2 + b_2, a_3 + b_3, ...) \end{equation}

And the same goes for all other operations.

$$a - b = (a_1 - b_1, a_2 - b_2, a_3 - b_3, ...)$$ $$a * b = (a_1 * b_1, a_2 * b_2, a_3 * b_3, ...)$$

Subsets of the hyperreal numbers

Hyperreal numbers can be divided in three main subsets

Infinitesimal numbers, such that $\forall x \in \mathbb{R},\ |\varepsilon| < x$
Finite numbers, which have a norm smaller than some real number : $\exists x \in \mathbb{R},\ |y| < |x|$
Infinite numbers, which have a norm larger larger than any real number : $\forall x \in \mathbb{R},\ |x| < |y|$

The canonical example of an infinitesimal number is the sequence $x_n = n^{-1}$. As it is strictly decreasing to $0$, it is easy to prove that for any strictly positive real number $y$, we have

$$\forall y > 0. \exists N.\ \forall n > N.\ |x_n| < y_n$$

But since the sequence is never $0$, it will always be superior to the sequence $y_n = 0$.

Conversely, the canonical infinite number is the sequence $x_n = n$, which

Last updated : 2020-03-10 10:08:33

Tags : analysis