Robinson arithmetic

In mathematics, Robinson arithmetic, or Q, is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out in R. M. Robinson (1950). Q is essentially PA without the axiom schema of induction. Q is weaker than PA but it has the same language, both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.

The background logic of Q is first-order logic with identity, denoted by infix '='. The individuals, called natural numbers, are members of a set called N with a distinguished member 0, called zero. There are three operations over N:

The following axioms for Q are Q1–Q7 in Burgess (2005: 56), and are also the first seven axioms of first-order arithmetic. Variables not bound by an existential quantifier are bound by an implicit universal quantifier.

The axioms in Robinson (1950) are (1)–(13) in Mendelson (1997: 201). The first 6 of Robinson's 13 axioms are required only when, unlike here, the background logic does not include identity. Machover (1996: 256–57) dispenses with axiom (3).

The usual strict total order on N, "less than" (denoted by "<"), can be defined in terms of addition via the rule x < y ↔ ∃z (Sz + x = y) (Burgess 2005:230, fn. 24).

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