Does Sigma(n)-KP, i.e. Kripke-Platek set theory with Sigma(n)-collection and Sigma(n)-separation have Sigma(n)-replacement.



A question on collection and replacement in Sigma(n)-KP
Yes. If we have a set $A$ and for every $a\in A$ there is a unique $b$ such that $\varphi(a,b,z)$, where $\varphi$ has complexity $\Sigma\_n$, then by $\Sigma\_n$ collecton there is a set $B$ such that every $a\in A$ has a $b\in B$ with $\varphi(a,b,z)$. But perhaps there is extra stuff in $B$, so this set may be too large to verify the desired instance of $\Sigma\_n$-replacement. But we may apply $\Sigma\_n$-separation to form the set $\{b\in B\mid \exists a\in A\ \varphi(a,b,z)\}$, which does fulfill the desired instance of $\Sigma\_n$-replacement. 


