linguistic:paraphrasing
Formal statement

In the formal language of the Zermelo–Fraenkel axioms, the axiom is as follows:

∃ I (∃ o (o ∈ I ∧ ¬ ∃ n n ∈ o) ∧ ∀ x (x ∈ I ⇒ ∃ y (y ∈ I ∧ ∀ a (a ∈ y ⇔ (a ∈ x ∨ a = x))))) {\displaystyle \exists I\ (\exists o\ (o\in I\ \land \ \lnot \exists n\ \ n\in o)\ \land \ \forall x\ (x\in I\Rightarrow \exists y\ (y\in I\ \land \ \forall a\ (a\in y\Leftrightarrow (a\in x\ \lor \ a=x)))))}

There exists a set 𝐼 (the set that is postulated to be infinite) such that the empty set is an element of it and for every element x {\displaystyle x} of 𝐼, there exists an element y {\displaystyle y} of 𝐼 consisting of just the elements of x {\displaystyle x} and x {\displaystyle x} itself.

This formula can be abbreviated as:

∃ I (∅ ∈ I ∧ ∀ x (x ∈ I ⇒ (x ∪ { x }) ∈ I)) {\displaystyle \exists I\,(\varnothing \in I\,\land \,\forall x\,(x\in I\Rightarrow \,(x\cup \{x\})\in I))}

Some mathematicians may call a set built this way an inductive set.