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# Meta-singular numbers

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

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 $\mathbb{R}\rightarrow\mathbb{R}$ 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 $a=\operatorname{xlim}(\{r(a)\}\times^{{\mathsf{FCD}}}x)$ 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 $r(\dots r(a)\dots)$ 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 this rough draft on my site (in PDF format).

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