On initial segments of degrees of constructibility

V. G. Kanovei

Matematičeskie zametki, Tome 17 (1975) no. 6, pp. 939-946

Résumé

Let $\mathfrak{M}$ be a fixed countable standard transitive model of $ZF+V=L$. We consider the structure Mod of degrees of constructibility of real numbers x with respect to $\mathfrak{M}$ such that $\mathfrak{M}$ (x) is a model. An initial segment $Q\subseteq\operatorname{Mod}$ is called realizable if some extension of $\mathfrak{M}$ with the same ordinals contains exclusively the degrees of constructibility of real numbers from $Q$ (and is a model of $ZFC$). We prove the following: if $Q$ is a realizable initial segment, then $\exists\,x\ [\forall\,y\ [x\in\operatorname{Mod}\&[y\in Q\to y<x]]\&\forall\,z\ \exists\,y\ [z<x\to y\in Q\&\ {\sim}[y<z]]]$.