On a forcing model for non-standard arithmetic

Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions
Covering semantics
For a coverage Cov the forcing clauses read as follows:
p −φ ∧ ψ ⇔ p −φ and p −ψ
p −φ ∨ ψ ⇔ {q ≤ p : q −φ or q −ψ} ∈ Cov(p)
p −φ → ψ ⇔ for all q ≤ p, q −φ implies q −ψ
p −∃xφ(x) ⇔ {q ≤ p : there exists an a such that q −φ(a)}
∈ Cov(p)
p −∀xφ(x) ⇔ for all q ≤ p and for all a, q −φ(a)
Warning: For a general coverage it is no longer the case that
classical logic is forced! But the laws of intuitionistic logic are.
Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic