On a forcing model for non-standard arithmetic

Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions
The forcing model
Let P be the preorder consisting of infinite subsets of N, preordered
by:
p ≤ q ⇔ ∃n ∀k ≥ n (k ∈ p → k ∈ q),
and let E = Sh(P, ¬¬). The object of truth values in E is
Ω(p) = {closed sieves on p},
where a sieve S is closed, if S dense below q implies q ∈ S. The
power set PN = Ω N is given by
Ω N (p) = {X : N → Ω(p)}.
Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic