These papers extend some points of the theory. The reading and understanding of these are not a prerequisite to a use of the theory.
This paper is about model theory and shows that the addition of the specific axioms do not enable to deduce any contradiction within set theory (unless it was already there).
The introduction of extra axioms allows for the construction of strange new objects. The ones that satisfy the usual properties of analysis are called internal. The others are external and are avoided when studying the traditional corpus of analysis.
An article which studies in depth the concept of external objects.
A simple article that shows how strange external functions can look!
Utrasmall numbers are a consequence of extra axioms but, like all axioms of set theory, they are about sets and not numbers. This paper shows the link between properties of sets and properties of numbers.