Meta-singular numbers

Not quite clear exposition and definition. Need to be more exact.

In http://www.mathematics21.org/algebraic-general-topology.htmlmy book I defined (generalized) limit of any (e.g. non-continuous) function.

But applying it, to say, a differential equation (replacing the limit in the definition of derivative with my generalized limit) does not work because the components of the equation may be of different types (for example generalized limit of a function $ℝ\to ℝ$ is not a real number).

To overcome this shortcoming I propose what I call meta-singular numbers.

[TODO: introduce the term singularity level above for a given number system.]

Let $a$ is a generalized limit. I will denote $r(a)$ such number (or generalized limit of a lower rank) that $a=\mathrm{xlim}(\{r(a)\}{\times }^{\mathrm{𝖥𝖢𝖣}}x)$ where $x$ is a filter (that is $a$ is a limit of a constant function), if such $r(a)$ exists. I will call reduced limit repeated applying $r(\mathrm{\dots }r(a)\mathrm{\dots })$ to a generalized limit $a$.

This definition is for now all I know about meta-singular numbers. It’s yet needed to formulate and prove some theorems.

See also http://www.mathematics21.org/binaries/reduced-limit.pdfthis rough draft on my site (in PDF format).