Corollary 3.34. Forcing with $Fn (ℵ_{1} \times ℵ_{3}, 2, ℵ_{1})$

(a)
Does not change $P (ω)$ .
(b)
Therefore preserves $ℵ_{1}$ (see Example Sheet 1 and the relation between codes for countable well-orders and preserving $ℵ_{1}$ ).