Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 4 / c

The axiom of infinity fails. First prove $L_n=V_n$ for every finite ordinal $n$. The base stage is empty in both hierarchies. If $L_n=V_n$, that stage is a finite set. Every one of its subsets is definable over it with finitely many parameters, by a disjunction of equalities to the selected elements; the empty subset is defined by a false formula. HenceInduction and the limit-stage definitions givethe set of all hereditarily finite sets. Every member of this set is finite externally.

If this transitive structure satisfied the axiom of infinity, it would contain an inductive set $I$: zero belongs to $I$, and whenever $x\in I$, its successor $x\cup \{x\}$ belongs to $I$. These operations are the actual empty set and actual set successor, since they are absolute for this transitive structure and all relevant finite-rank sets lie in it. External induction would then place every finite ordinal in $I$, contradicting the finiteness of $I$.

Every individual finite ordinal does belong to $L_{\omega }$, but the set $\omega$ of all finite ordinals does not. The distinction between containing each finite ordinal and containing one inductive set is precisely the issue.