On a forcing model for non-standard arithmetic

Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions
The ultrafilter
So suppose X ∈ Ω N (p), i.e., X is a function N → Ω(p).
The ultrafilter is given by: p −X ∈ U iff
{q ≤ p : q ∈ X (n) for all n ∈ q} is dense below p.
One easily verifies that E believes this to be a filter, which does
not contain the finite sets. So the main issue is to show that it is
in fact an ultrafilter.
Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic