  It was established by Jensen in 1970 that there is a generic extension $L[a]$
of the constructible universe $L$ by a real $a\not\in L$ such that $a$ is
$\varDelta^1_3$ 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 $\Diamond$) 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 $\varPi^1_2$ singleton as a
class-forcing extension of a model of ZFC$^-$ plus $V=L$.

Logic