On slow minimal reals I

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

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