Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 16G / Solution

The cumulative hierarchy is defined by transfinite recursion:The rank of a set is $\mathrm{rk}(x)={\sup }_{y\in x}(\mathrm{rk}(y)+1)$, equivalently the least $\alpha$ such that $x\subseteq V_{\alpha }$. The standard hierarchy properties give $\mathrm{rk}(V_{\alpha })=\alpha$, so every ordinal occurs as a set rank; the ordinal $\alpha$ itself is another example of rank $\alpha$.

Let $x$ be transitive of rank $\alpha$, and fix $\beta <\alpha$. The supremum formula gives an element of $x$ whose rank is at least $\beta$. Choose such an element $y$ of least possible rank. If $\mathrm{rk}(y)>\beta$, the same supremum formula gives $z\in y$ with rank at least $\beta$. Transitivity gives $z\in x$, and membership strictly decreases rank, contradicting the choice of $y$. Thus $x$ contains an element of every rank below $\alpha$.

A nonempty finite set has rank the maximum of finitely many successor ordinals, hence a successor ordinal; the empty set has rank zero. Conversely, for any $\beta$, the singleton $\{V_{\beta }\}$ has rank $\beta +1$. ThusA finite transitive set of rank $\alpha$ contains, by the proved property, distinct elements of every rank $\beta <\alpha$. Finiteness therefore forces $\alpha <\omega$. Every finite ordinal is itself a finite transitive set of that rank. Consequently