as i understand it, although you need not actually take as axiomatic the existence of the null set, it is often done.

once you have the null set, you can form the set of all subsets of the null set (the power set). since the null set HAS no subsets, the power set just

contains one element, the null set itself. now we have two sets. given any two sets, we can form their union,

which in this case, is typically called the natural number 1 (and the power set of the null set is called 0).

continuing, we can define 2 as 1 U {1}. then the axiom of infinity says there exists a set: {1,2,3,4,......}.

this is equivalent to saying that the natural numbers form a set (that is, that the natural numbers are a well-defined set

formed by well-defined set operations from the one set we are sure exists: the null set).

what your version of the axiom of infinity actually says, in ordinary english, is: "there exists an inductive set".

the fact that

establishes the "base case", and the last part establishes the "inductive step"

if

then

(in a usual definition of the natural numbers, this is saying:

if n is in X, then so is n+1).

it is quite possible to have a model of set theory without this axiom (in which every set is finite). which although logically satisfactory, leaves us in the position of not being able to use constructions we take for granted in mathematics, such as defining real numbers as convergent sequences of rational numbers.