Definable minimal collapse functions at arbitrary projective levels

Abstract

Using a non-Laver modification of Uri Abraham's minimal ${\Delta }_3^1$ collapse function, we define a generic extension $L[a]$ by a real $a$, in which, for a given $n\ge 3$, $\{a\}$ is a lightface ${\Pi }_n^1$ singleton, $a$ effectively codes a cofinal map $\omega \to {\omega }_1^L$ minimal over $L$, while every ${\Sigma }_n^1$ set $X\subseteq \omega$ is still constructible.

30 pages

References in corpus (2)

• Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
• Definable ${\mathsf{E}}_0$ classes at arbitrary projective levels

Cited by in corpus (2)

• A model in which the Separation principle holds for a given effective projective Sigma-class
• Indiscernible pairs of countable sets of reals at a given projective level