On slow minimal reals I

Abstract

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