Kripke Platek set theory has collection instead of replacement, and it is a weakening of KP if one has replacement instead of collection. Call KP minus collection plus replacement KF for *Kripke Fraenkel*. Is KF weaker than KP in that some transfinite recursion can be done by KP which cannot be done by KF? If so, is that a difference inherited by strengthened theories as $\Sigma \_{n} KP$ and $\Sigma \_{n} KF$ for $n>1$?



Transfinite recursion, collection and replacement in KP and KF
The main deficit of KF over KP is that one cannot prove that the set of formulae equivalent to a $\Sigma\_1$ or $\Pi\_1$ formula is closed under bounded quantification. (And this persists up through the $KF\_n/KP\_n$ you define.)


Therefore recursions involving schemes defined by a formula of the form $\forall u \in v \varphi$ cannot in general be effected in KF.


At the level n=1: Mathias in: ``Weak Systems of Gandy, Jensen and Devlin'', (Trends in Mathematics, Ed. Bagaria, Todorcevic, Birkhäuser Press) states that methods of Zarach should be sufficient to give an example showing that $\Delta\_0$-collection cannot be substituted
by $\Delta\_0$-replacement, but does not give an example (and I don't have one).


At levels > 1: See Gitman, Johnstone, Hamkins ``What is the theory ZFC without Power Set".
(Archive for Math.Logic, 2012) where they give examples (some derived from Zarach) of failures of even low levels of $\Sigma\_n$-Collection even in models of full $\Sigma\_\omega$-Replacement. They observe that one can inductively define in one such model a family of bijective functions $f:\omega \longrightarrow \aleph\_k$ for $k\leq n$, but not altogether for all $n<\omega$. Thus strengthening the replacement scheme does not help.


