[1] It is usually denoted Q. Q is almost[clarification needed] PA without the axiom schema of mathematical induction.
Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.
The first 6 of Robinson's 13 axioms are required only when, unlike here, the background logic does not include identity.
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).
Equivalently, we get a definitional conservative extension of Q by taking "<" as primitive and adding this rule as an eighth axiom; this system is termed "Robinson arithmetic R" in Boolos, Burgess & Jeffrey (2002, Sec 16.4).
Shoenfield's system also appears in Boolos, Burgess & Jeffrey (2002, Sec 16.2), where it is called the "minimal arithmetic" (also denoted by Q).
However, unlike Peano arithmetic, Tennenbaum's theorem does not apply to Q, and it has computable non-standard models.
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.
Q is a finitely axiomatized first-order theory that is considerably weaker than Peano arithmetic (PA), and whose axioms contain only one existential quantifier.
[6][7][8] 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.
[9][10][11] 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).