Models of arithmetic

A subject the papers are about. The loosest grouping, and the one to reach for last.

Arithmetic in the models that are not the intended one: initial segments, inconsistent models, and weakened induction.

There are three distinct ways to leave the standard model, and the set holds all of them rather than one.

*More models than intended.* Kochen and then Frayne, Morel and Scott supply the machinery -- ultraproducts and reduced products -- and the pairing is the fastest way into the set: Frayne's introduction records that Scott's work was stimulated by Kochen, 'who wanted to extend Skolem's method for obtaining models of arithmetic to more general situations'. Kaye's textbook and Kossak and Schmerl on the structure of models are where that programme ended up; Ikeda on nonstandard models definable inside models of PA is the reflexive case.

*Weaker theories.* Drop multiplication and arithmetic becomes decidable: Haase twice on Presburger, the survival guide and the complexity subclasses, with Krynicki on arithmetic in finite models. These are not failures to reach the standard model but deliberate retreats from it, and they are the members most likely to be actually used.

*Inconsistent models.* Mortensen's inconsistent mathematics and Priest in two parts, finite models then the general case. Arithmetic in which contradictions are true and contained -- the furthest departure here, and the one that makes the set's title a real claim rather than a synonym for nonstandard.

Hajek and Pudlak is the reference behind all three. Wang's axiomatization paper is shared with `the-axioms-before-peano`, where it does different work: here it is background on what the axioms are, there it is the evidence that Grassmann had them first. Diaconescu approaches the same territory categorically and stands somewhat alone.