Jensen ${\Delta }_3^1$ reals by means of ZFC${}^{-}$ or second order Peano arithmetic

It was established by Jensen in 1970 that there is a generic extension $L[a]$ of the constructible universe $L$ by a real $a\in ̸L$ such that $a$ is ${\Delta }_3^1$ in $L[a]$. Jensen’s forcing construction has found a number of applications in modern set theory. A problem has been recently discussed whether Jensen’s construction can be reproduced entirely within second order Peano arithmetic or equivalently FC${}^{-}$ (minus the Power Set axiom). The obstacle is that the proof of the key CCC property (whether by Jensen’s original argument or a later proof using $\mathrm{◊}$) essentially involves countable elementary submodels of $L_{{\omega }_2}$, which is way beyond ZFC${}^{-}$. We show how to circumwent this difficulty by means of killing only definable antichains in the course of a Jensen-like transfinite construction of the forcing, and then define a model with a minimal ${\Pi }_2^1$ singleton as a class-forcing extension of a model of ZFC${}^{-}$ plus $V=L$.