  We prove that, over Kripke-Platek set theory with infinity (KP), transfinite
induction along the ordinal ${\epsilon}_{\Omega+1}$ is equivalent to the schema
asserting the soundness of KP, where $\Omega$ denotes the supremum of all
ordinals in the universe; this is analogous to the result that, over Peano
arithmetic (PA), transfinite induction along ${\epsilon}_0$ is equivalent to
the schema asserting the soundness of PA. In the proof we need to code
infinitary proofs within KP, and it is done by using partial recursive set
functions. This result can be generalised to KP + $\Gamma$-separation +
$\Gamma$-collection where $\Gamma$ is any given syntactic complexity, but not
to ZF.
