On slow minimal reals I

TL;DR

This paper constructs a proper forcing that adds a minimal real, which is eventually different from all old reals, in a slowly growing product space, answering a question by Harrington.

Contribution

It introduces a proper forcing notion that adds a minimal real with specific properties in a slowly growing product space, addressing a question in set theory.

Findings

Existence of a proper forcing adding a minimal real

The minimal real is eventually different from all old reals

The construction works for a slowly growing sequence

Abstract

Answering a question of Harrington, we show that there exists a proper forcing notion, which adds a minimal real $\eta \in \prod_{i<\omega} n^*_i$, which is eventually different from any old real in $\prod_{i<\omega} n^*_i$, where the sequence $\langle n^*_i \mid i<\omega \rangle$ grows slowly.