On a forcing model for non-standard arithmetic

Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions
Topos-theoretic perspective, continued
A presheaf is a sheaf (relative to a coverage Cov) if it satisfies:
Sheaf axiom
For every S ∈ Cov(p) and family of elements {xq ∈ X (q) : q ∈ S}
that is compatible (i.e. satisfies p ≤ q ∈ S ⇒ xq · p = xp), there is
a unique element x ∈ X (p) such that x · q = xq for all q ∈ S.
Sheaves also form a topos Sh(P, Cov). The internal logic of
categories of sheaves is precisely covering semantics. The internal
logic of Sh(P, ¬¬) is classical forcing.
Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic