Past exam of the axiomatic set theory course of the University of Oxford / 2011 / 2 / c

There is a bounded definition of omega. Transitivity has the bounded definition $\mathrm{Trans}(x)\equiv \forall u\in x\forall v\in u(v\in x)$. An ordinal is a transitive set of transitive sets on which membership is a strict total order; this too has a bounded formula in set theory, since ZF supplies well-foundedness by the Axiom of foundation.

Define the bounded predicateFor an ordinal, this says that it is a nonzero limit ordinal. It follows in ZF thatsince $\omega$ is the least nonzero limit ordinal. The right side is a bounded formula in set theory; consequently it is absolute between the specified transitive models. This also shows directly that those models have the same $\omega$.

Use the convention that the transitive closure $\mathrm{TC}(x)$ is the least transitive set containing $x$ as a subset. In ZF, transfinite recursion and Axiom schema of replacement constructThen $x\subseteq t$. If $a\in b\in t$, some $n$ has $b\in x_n$, whence $a\in x_{n+1}\subseteq t$. Thus $t$ is transitive. If another transitive set $s$ contains $x$, induction gives $x_n\subseteq s$ for every $n$, and hence $t\subseteq s$. This proves existence and uniqueness of the transitive closure.

Let $W(w)$ be the bounded definition of omega above. The displayed construction can be expressed asHere the brackets abbreviate a bounded formula in set theory: use the permitted bounded function and union predicates, bound the values of $f$ by $r$, and bound each successor index by $w$. Ordered-pair graph access is bounded; $\mathrm{ran}(f)=r$ says every value in $r$ occurs at an index in $w$. Any auxiliary sets used for the graph coding may be put in the initial existential block. Thus this is a Sigma-one formula in set theory defining $y=\mathrm{TC}(x)$.

Denote this formula by $T(x,y)$. Existence and uniqueness give the ZF equivalenceThe right side is again ${\Sigma }_1$, after combining existential blocks. Consequently $T$ is a Delta-one formula modulo ZF, so Delta-one absoluteness applies. Therefore transitive closure is Delta-one definable and is ZF-absolute. If closure is instead required to contain $x$ as an element, apply exactly this construction to $\{x\}$; the absoluteness conclusion is unchanged.