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# Robinson arithmetic

In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950. [1]

1. 52 relations: Addition, Axiom, Axiom of adjunction, Axiom schema, Binary operation, Cambridge University Press, Cardinality, Complete theory, Computable function, Converse (logic), Domain of a function, Empty set, Equality (mathematics), Existential quantification, Extensionality, Gödel numbering, Gödel's incompleteness theorems, General set theory, Gentzen's consistency proof, Infinite set, Injective function, Interpretation (logic), Journal of Symbolic Logic, List of first-order theories, Mathematical induction, Mathematics, Multiplication, Natural number, Non-standard model of arithmetic, Operation (mathematics), Oxford University Press, Peano axioms, Polish notation, Polynomial, Presburger arithmetic, Princeton University Press, Raphael M. Robinson, Recursive definition, Second-order arithmetic, Set (mathematics), Set theory, Set-theoretic definition of natural numbers, Skolem arithmetic, Springer Science+Business Media, Successor function, Tennenbaum's theorem, Total order, Unary operation, Universal quantification, Variable (mathematics), ... Expand index (2 more) »

2. Formal theories of arithmetic

Addition (usually signified by the plus symbol) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

## Axiom

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w.

## Axiom schema

In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.

## Binary operation

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element.

## Cambridge University Press

Cambridge University Press is the university press of the University of Cambridge.

## Cardinality

In mathematics, the cardinality of a set is a measure of the number of elements of the set.

## 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.

## Computable function

Computable functions are the basic objects of study in computability theory.

## Converse (logic)

In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements.

## Domain of a function

In mathematics, the domain of a function is the set of inputs accepted by the function.

## Empty set

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.

## Equality (mathematics)

In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object.

## Existential quantification

In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".

## Extensionality

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties.

## Gödel numbering

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number.

## Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories.

## General set theory

General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.

## Gentzen's consistency proof

Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936.

## Infinite set

In set theory, an infinite set is a set that is not a finite set.

## Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.

## Interpretation (logic)

An interpretation is an assignment of meaning to the symbols of a formal language.

## Journal of Symbolic Logic

The Journal of Symbolic Logic is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic.

## List of first-order theories

In first-order logic, a first-order theory is given by a set of axioms in some language.

## Mathematical induction

Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots&thinsp; all hold.

## Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

## Multiplication

Multiplication (often denoted by the cross symbol, by the mid-line dot operator, by juxtaposition, or, on computers, by an asterisk) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division.

## Natural number

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0.

## 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. Robinson arithmetic and non-standard model of arithmetic are formal theories of arithmetic.

## Operation (mathematics)

In mathematics, an operation is a function which takes zero or more input values (also called "operands" or "arguments") to a well-defined output value.

## Oxford University Press

Oxford University Press (OUP) is the publishing house of the University of Oxford.

## Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. Robinson arithmetic and Peano axioms are formal theories of arithmetic.

## Polish notation

Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators precede their operands, in contrast to the more common infix notation, in which operators are placed between operands, as well as reverse Polish notation (RPN), in which operators follow their operands.

## Polynomial

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

## Presburger arithmetic

Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. Robinson arithmetic and Presburger arithmetic are formal theories of arithmetic.

## Princeton University Press

Princeton University Press is an independent publisher with close connections to Princeton University.

## Raphael M. Robinson

Raphael Mitchel Robinson (November 2, 1911 – January 27, 1995) was an American mathematician.

## Recursive definition

In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff).

## Second-order arithmetic

In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. Robinson arithmetic and second-order arithmetic are formal theories of arithmetic.

## Set (mathematics)

In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

## Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.

## Set-theoretic definition of natural numbers

In set theory, several ways have been proposed to construct the natural numbers.

## Skolem arithmetic

In mathematical logic, Skolem arithmetic is the first-order theory of the natural numbers with multiplication, named in honor of Thoralf Skolem. Robinson arithmetic and Skolem arithmetic are formal theories of arithmetic.

Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Successor function

In mathematics, the successor function or successor operation sends a natural number to the next one.

## Tennenbaum's theorem

Tennenbaum's theorem, named for Stanley Tennenbaum who presented the theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).

## Total order

In mathematics, a total order or linear order is a partial order in which any two elements are comparable.

## Unary operation

In mathematics, a unary operation is an operation with only one operand, i.e. a single input.

## Universal quantification

In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", or "for any".

## Variable (mathematics)

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object.

## Zermelo set theory

Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG).

## 0

0 (zero) is a number representing an empty quantity.