Definable minimal collapse functions at arbitrary projective levels

TL;DR

This paper introduces a new method to construct generic extensions where a real codes a minimal cofinal map over L, maintaining definability at arbitrary projective levels.

Contribution

It develops a non-Laver modification of Abraham's collapse function to produce definable minimal collapse functions at any projective level.

Findings

Constructs a generic extension L[a] with a real coding a minimal cofinal map.

Ensures all Σ^1_n sets remain constructible in the extension.

Provides a definable minimal collapse function at arbitrary projective levels.

Abstract

Using a non-Laver modification of Uri Abraham's minimal $\varDelta^1_3$ collapse function, we define a generic extension $L[a]$ by a real $a$, in which, for a given $n\ge3$, $\{a\}$ is a lightface $\varPi^1_n$ singleton, $a$ effectively codes a cofinal map $\omega\to\omega_1^L$ minimal over $L$, while every $\varSigma^1_n$ set $X\subseteq\omega$ is still constructible.