No,
the empty is the set with no elements. It is defined by the empty set axiom: ∃
x∀
y(
y ∉
x).
You can define the natural numbers (including 0) using any set that satisfies
Peano's axioms. So, it is
possible (but not necessary) to represent 0 by the empty set. However, the axiom of infinity has nothing to do with
dividing by 0.
The axiom of infinity states the existence of a set x.
It explicitly states that the empty set is in x (∅ ∈ x). It then says: and (∧) for all y (∀y) if y is in x (y ∈ x) then (⇒) the union of y and a set containing y (y ∪ {y}) is in x (∈ x).
So, we can start with the element ∅ and apply the rule:-
∅ ∪ {∅} = {∅}
{∅} ∪ {{∅}} = {∅, {∅}}
{∅, {∅}} ∪ {{∅, {∅}}} = {∅, {∅}, {∅, {∅}}}
So
x ⊇ {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}},...}
So x is a set that contains a set that satisfies Peano's axioms, which in turn can be identified with the set of natural numbers {0, 1, 2, 3,...}
So, we've created all the natural numbers form nothing - isn't maths fun!