TITLE: What are the consequences if Axiom of Infinity is negated?
QUESTION [13 upvotes]: 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

REPLY [2 votes]: I'm not certain why this isn't in the previous answer, but it seems worth mentioning that in first-order PA you cannot restrict the models to the specific one you want (the natural numbers 0, 1, 2, ...).  Put another way, the same axioms of PA are satisfied by, for one example, a model where you have 0, 1, 2, ..., a, b, c, d, ..., where the latter set of "numbers" are an infinite distance from the former (standard) set.  First-order PA is essentially too weak to define "natural number" in the restrictive manner that is typically intended.
The point I wanted to make is that your axiomatization does not change this fact.  I'm not certain off-hand whether it is consistent with the definitions for + and *, but if it is, it still allows for (can be modeled by) an infinite number of objects (e.g. 0, 1, 2, ..., a, b, c, inf.).