On a forcing model for non-standard arithmetic

Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions
Coverages
This can be generalised using the notion of a coverage.
A sieve on p is a downwards closed subset of ↓ p. A coverage Cov
is a function that assigns to every p ∈ P a collection Cov(p) of
sieves on p, in such a way that three axioms are satisfied:
1 ↓ p ∈ Cov(p);
2 if q ≤ p and S ∈ Cov(p), then S∩ ↓ q ∈ Cov(q);
3 if S ∈ Cov(p) and Rq ∈ Cov(q) for every q ∈ S, then
∪q∈SRq ∈ Cov(p).
Example: the ¬¬-coverage.
Cov(p) = {S ⊆↓ p : S is downwards closed and dense below p}.
Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic