[jakef]

The natural numbers are each finite. In set theory, the standard way they are defined is that they can be constructed by assuming the existence of the empty set (or 0) and assuming that if you "insert a set into itself" the result will be a set. (So they can be thought of as {} = 0, {{}} = 1, {{}, {{}}} = 2, etc.).

The natural numbers are simply the (smallest) set that contains the empty set and is closed under this "insert a set into itself" operation (successor). It only contains finite sets since the successor operation will never turn an finite set into an infinite set.

The existence of this set of ALL natural numbers (an infinite object) relies on an axiom, the axiom of infinity.

There is no transfinite natural number.