Robinson arithmetic

Axioms

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.

1. Sx ≠ 0
2. 0 is not the successor of any number.
3. (Sx = Sy) → x = y
4. If the successor of x is identical to the successor of y, then x and y are identical. (1) and (2) yield the minimum of facts about N (it is an infinite set bounded by 0) and S (it is an injective function whose domain is N) needed for non-triviality. The converse of (2) follows from the properties of identity.
5. y=0 ∨ ∃x (Sx = y)
6. Every number is either 0 or the successor of some number. The axiom schema of mathematical induction present in arithmetics stronger than Q turns this axiom into a theorem.
7. x + 0 = x
8. x + Sy = S(x + y)
9. (4) and (5) are the recursive definition of addition.
10. x·0 = 0
11. x·Sy = (x·y) + x
12. (6) and (7) are the recursive definition of multiplication.

Variant axiomatizations

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).

Taking "<" as primitive requires adding four axioms to (1)–(7) above:

Another axiomatization using < may be found in Shoenfield (1987: 22).

Metamathematics

On the metamathematics of Q, see Boolos et al. (2002: chpt. 14), Tarski, Mostowski, and Robinson (1953), Smullyan (1991), Mendelson (1997: 201–03), and Burgess (2005: §§1.5a, 2.2). The intended interpretation of Q is the natural numbers and their usual arithmetic in which addition and multiplication have their customary meaning, identity is equality, Sx = x + 1, and 0 is the natural number zero.

Q, like Peano arithmetic, has nonstandard models of all infinite cardinalities. However, unlike Peano arithmetic, Tennenbaum's theorem does not apply to Q, and it has computable non-standard models. For instance, there is a computable model of Q consisting of integer-coefficient polynomials with positive leading coefficient, plus the zero polynomial, with their usual arithmetic.

The defining characteristic of Q is the absence of the axiom scheme of induction. Hence it is often possible to prove in Q every specific instance of a fact about the natural numbers, but not the associated general theorem. For example, 5 + 7 = 7 + 5 is provable in Q, but the general statement x + y = y + x is not. Similarly, one cannot prove that Sx ≠ x (Burgess 2005:56).

Q is interpretable in a fragment of Zermelo's axiomatic set theory, consisting of extensionality, existence of the empty set, and the axiom of adjunction. This theory is S' in Tarski et al. (1953: 34) and ST in Burgess (2005: 90–91; 223). See general set theory for more details.

Q is fascinating because it is a finitely axiomatized first-order theory that is considerably weaker than Peano arithmetic (PA), and whose axioms contain only one existential quantifier, yet like PA is incomplete and incompletable in the sense of Gödel's incompleteness theorems, and essentially undecidable. Robinson (1950) derived the Q axioms (1)–(7) above by noting just what PA axioms are required to prove (Mendelson 1997: Th. 3.24) that every computable function is representable in PA. The only use this proof makes of the PA axiom schema of induction is to prove a statement that is axiom (3) above, and so, all computable functions are representable in Q (Mendelson 1997: Th. 3.33, Rautenberg 2010: 246). The conclusion of Gödel's second incompleteness theorem also holds for Q: no consistent recursively axiomatized extension of Q can prove its own consistency, even if we additionally restrict Gödel numbers of proofs to a definable cut (Bezboruah and Shepherdson 1976; Pudlák 1985; Hájek & Pudlák 1993:387).

The first incompleteness theorem applies only to axiomatic systems defining sufficient arithmetic to carry out the necessary coding constructions (of which Gödel numbering forms a part). The axioms of Q were chosen specifically to ensure they are strong enough for this purpose. Thus the usual proof of the first incompleteness theorem can be used to show that Q is incomplete and undecidable. This indicates that the incompleteness and undecidability of PA cannot be blamed on the only aspect of PA differentiating it from Q, namely the axiom schema of induction.

Gödel's theorems do not hold when any one of the seven axioms above is dropped. These fragments of Q remain undecidable, but they are no longer essentially undecidable: they have consistent decidable extensions, as well as uninteresting models (i.e., models which are not end-extensions of the standard natural numbers).