We answer some problems set by Priest in  and , in particular refuting Priest’s Conjecture that all LP-models of Th(ℕ) essentially arise via congruence relations on classical models of Th(ℕ). We also show that the analogue of Priest’s Conjecture for IΔ0 + Exp implies the existence of truth definitions for intervals [0,a] ⊂e M ⊨ IΔ0 + Exp in any cut [0,a] ⊂e K ⊆e M closed under successor and multiplication.
"On LP-models of arithmetic." J. Symbolic Logic 73 (1) 212 - 226, March 2008. https://doi.org/10.2178/jsl/1208358750