Observability and Mathematics

2 Several definitions and statements of Mathematics with Observers

For references, see the book [8].

Sets W_n and W_n-observers

We call ${\mathit{W}}_{\mathit{n}}$ the set of all decimal fractions, such that there are at most n digits in the integer part and n digits in the decimal part of the fraction. Visually, an element in ${\mathit{W}}_{\mathit{n}}$ looks like

$\begin{array}{r}\pm \underset{\mathit{n}}{\underbrace{\begin{array}{ccc}\text{\_}& \dots & \text{\_}\end{array}}}.\underset{\mathit{n}}{\underbrace{\begin{array}{ccc}\text{\_}& \dots & \text{\_}\end{array}}}\end{array}$

We get ${\mathit{W}}_{\mathit{k}}\subset {\mathit{W}}_{\mathit{n}}$ if .

We call a ${\mathit{W}}_{\mathit{n}}$-observer any system working within ${\mathit{W}}_{\mathit{n}}$.

The set of ${\mathit{W}}_{\mathit{n}}$-observers is a finite well-ordered system ordered by n, and a ${\mathit{W}}_{\mathit{n}}$-observer sees what and how any ${\mathit{W}}_{\mathit{k}}$-observer with is doing in ${\mathit{W}}_{\mathit{k}}$.

But a ${\mathit{W}}_{\mathit{k}}$-observer is unaware (or can only assume) the existence of ${\mathit{W}}_{\mathit{n}}$-observers with $\mathit{n}\mathrm{>}\mathit{k}$.

Let us note, for example, that a ${\mathit{W}}_2$-observer cannot see the full set ${\mathit{W}}_2$, and a ${\mathit{W}}_3$-observer sees what and ...