This definition is the only one I know of a limit. I never met anyone who would use another definition. Everything is built from this definition: derivative, integral, statistics, probability theory, calculus, numerical methods, ...

There's nothing stopping you from inventing new definitions, but then you kind of have to explain why they're useful. How exactly do you define the sum of all natural numbers (or equality)?

It looks like this theory doesn't prove any new theorems from set theory, so I'm not entirely sure how it'd affect the sum of natural numbers.

Low level theory kind of takes care of the construction of the integers and tells you that you can do whatever feels right with them. Abstract results were based from those intuitive feelings about integers. So, that long road was already constructed, it was just fixed a bit at the beginning.

It's like people were building everything in Python for centuries than they wanted everything to be build in assembly. So they just had to write Python in assembly.

Basically:

"God created the integers, everything else is the work of man"