What mathematics can be built with standard ZFC with Axiom of Infinity replaced with its negation?
Can the analysis be built? Is there special name for "ZFC without Infinity" set theory?
I also assume that using symbols like $+\infty$, $-\infty$ when dealing with properties of functions and their limits would still be possible even with Axim of Infinity negated (correct me if I am wrong).
UPDATE
In light of the answer by Andres Caicedo which suggested Peano arithmetic, I want to point out that Wikipedia says about Peano arithmetic "Axioms 1, 6, 7 and 8 imply that the set of natural numbers is infinite, because it contains at least the infinite subset". I do not know how this can be interpreted as having axiom of infinity but I am interested what if Peano arithmetic modified the following way:
- Added an axiom 5a:
There is a natural number $\infty$ which has no successor, for any natural number n $S(\infty)=n$ is false.
- Axiom 6 modified:
For every natural number n except $\infty$, S(n) is a natural number