On a forcing model for non-standard arithmetic

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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
Sheaf semantics A non-principal ultrafilter A forcing model for non-standard arithmetic Open questions Topos-theoretic perspective A presheaf on P is a functor P op → Sets. So it consists of a family of sets {X (p) : p ∈ P} together with a restriction operation assigning to every element x ∈ X (p) and q ≤ p an element x · q ∈ X (q). Presheaves form a topos PSh(P). Every topos has an “internal logic”: the internal logic of categories of presheaves is Kripke semantics (for intuitionistic logic). Benno van den Berg, TU Darmstadt, Germany On a forcing model for non-standard arithmetic
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On a forcing model for non-standard arithmetic

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