Theory of singularities using generalized limits
portonmath, 25 November 2013 (created 9 November 2013)
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This issue was raised in this blog post. You may comment on this blog post about your research and questions on the issues raised on this wiki.

Motto: Unravel the mystery of singularities.

Warning: It is a rough draft and may contain errors.

In my book Algebraic General Topology. Volume 1 I introduced the concept of generalized limit. (Generalized limit, unlike traditional limit, is defined for arbitrary values, even in singularity points.)

This wiki is intended to rigorously define (and research) singularities using values of generalized limits in equations. The trouble is that generalized limit is not a number (technically it is a set of funcoids) and can't be put into the same equation as a number without "type casting". It is possible to cast values (such as real numbers) into sets of funcoids to move them into the same "space" as generalized limits. But this does not work because there may be singularities "of level above" that is "singularities of singularities".

Thus there should be produced an infinite hierarchy of singularities: starting from numbers (such as real numbers), then plain singularities, then singularities of singularities, etc.

In this rough draft article I attempted to introduce "metasingular numbers" to construct an infinite hierarchy of singularities and overcome this problem. The problem appeared to be difficult however.

So this wiki is to define metasingular numbers exactly and construct the infinite hierarchy of singularities.

The overall idea
Attempted ways to define singularity level above
Using plain funcoids
Singularities funcoids: some special cases
Singularities funcoids: special cases proof attempts
Using generalized funcoids
Galufuncoids
Functional galufuncoids
More on galufuncoids
Functions with meta-singular numbers as arguments
On differential equations
Special case of general relativity
Cheap way