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

Abstract

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 $V=L$.