Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 6 / iii

Use the bounded formulaIt expresses $x\subseteq y$ with no unbounded quantifier. For parameters $x,y\in U_1$, transitivity puts every member of $x$ in $U_1$ already, and membership in $y$ has the same meaning in both classes. Thus the formula is ${\Delta }_0$ and is absolute in both directions, in particular upward absolute.

Equality with the union is expressed byThe first conjunct says that every member of $x$ lies in some member of $y$; the second gives the reverse inclusion. Every quantifier is bounded by a parameter or by an element of a previously bounded parameter. Therefore bounded absoluteness applies, proving full absoluteness and hence upward absoluteness of $x=\bigcup y$. This only compares the relation for existing parameters; it does not require either class to contain the union of every one of its sets.

A bounded expression for the successor relation isThe first two clauses put $y$ and all its elements into $x$, while the last forbids any additional member of $x$. Extensionality identifies this exactly with $x=y\cup \{y\}$. Since the formula is ${\Delta }_0$, it is absolute in both directions between the transitive classes and therefore upward absolute. This bounded description also justifies the successor computations used earlier in the rank-stage Replacement counterexample.