Concept

Robinson arithmetic

In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and 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: A unary operation called successor and denoted by prefix S; Two binary operations, addition and multiplication, denoted by infix + and ·, respectively. The following axioms for Q are Q1–Q7 in (cf. also the axioms of first-order arithmetic). Variables not bound by an existential quantifier are bound by an implicit universal quantifier. Sx ≠ 0 0 is not the successor of any number. (Sx = Sy) → x = y 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. y=0 ∨ ∃x (Sx = y) 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. x + 0 = x x + Sy = S(x + y) (4) and (5) are the recursive definition of addition. x·0 = 0 x·Sy = (x·y) + x (6) and (7) are the recursive definition of multiplication. The axioms in are (1)–(13) in . 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 "

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Second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted by Z2.
Non-standard model of arithmetic
In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, .... The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).
Complete theory
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence the theory contains the sentence or its negation but not both (that is, either or ). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem.
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