On a forcing model for non-standard arithmetic

Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions
Intuitionistic model 2
Let B be the powerset of N, ordered by inclusion, and consider the
ideal I of finite sets in N. I determines a coverage on B:
S ∈ Cov(c) ⇔ ∃i ∈ I , b1, . . . , bn ∈ S : b1 ∨ . . . ∨ bn ∨ i = c,
which we will also denote by I . In E = Sh(B, I ) the following
defines an filter on N:
p −X ∈ U ⇔ {q ≤ p : q ∈ X (n) for all n ∈ q} ∈ Cov(p).
Forcing clauses for the atomic formulas are as above and the
clauses for the interpretation of logical connectives are as in
covering semantics.
Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic