 Solution

ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2009/b/6/iii/c/solution

A bounded expression for the successor relation is

y \in x \land (\forall z \in y z \in x) \land (\forall z \in x (z \in y \lor z = y)) .

The 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

Δ_{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.

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