## Table of Contents

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) »

- Formal theories of arithmetic

## Addition

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

See Robinson arithmetic and Addition

## 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.

See Robinson arithmetic and Axiom

## Axiom of adjunction

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

See Robinson arithmetic and Axiom of adjunction

## Axiom schema

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

See Robinson arithmetic and Axiom schema

## Binary operation

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

See Robinson arithmetic and Binary operation

## Cambridge University Press

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

See Robinson arithmetic and Cambridge University Press

## Cardinality

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

See Robinson arithmetic and Cardinality

## 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.

See Robinson arithmetic and Complete theory

## Computable function

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

See Robinson arithmetic and Computable function

## Converse (logic)

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

See Robinson arithmetic and Converse (logic)

## Domain of a function

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

See Robinson arithmetic and Domain of a 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.

See Robinson arithmetic and Empty set

## 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.

See Robinson arithmetic and Equality (mathematics)

## 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".

See Robinson arithmetic and Existential quantification

## Extensionality

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

See Robinson arithmetic and Extensionality

## 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.

See Robinson arithmetic and Gödel numbering

## 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.

See Robinson arithmetic and Gödel's incompleteness theorems

## 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.

See Robinson arithmetic and General set theory

## Gentzen's consistency proof

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

See Robinson arithmetic and Gentzen's consistency proof

## Infinite set

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

See Robinson arithmetic and Infinite 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.

See Robinson arithmetic and Injective function

## Interpretation (logic)

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

See Robinson arithmetic and Interpretation (logic)

## Journal of Symbolic Logic

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

See Robinson arithmetic and Journal of 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.

See Robinson arithmetic and List of first-order theories

## 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  all hold.

See Robinson arithmetic and Mathematical induction

## 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.

See Robinson arithmetic and Mathematics

## 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.

See Robinson arithmetic and Multiplication

## Natural number

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

See Robinson arithmetic and Natural number

## 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.

See Robinson arithmetic and Non-standard model 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.

See Robinson arithmetic and Operation (mathematics)

## Oxford University Press

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

See Robinson arithmetic and Oxford University Press

## 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.

See Robinson arithmetic and Peano axioms

## 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.

See Robinson arithmetic and Polish notation

## 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.

See Robinson arithmetic and Polynomial

## 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.

See Robinson arithmetic and Presburger arithmetic

## Princeton University Press

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

See Robinson arithmetic and Princeton University Press

## Raphael M. Robinson

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

See Robinson arithmetic and Raphael M. Robinson

## 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).

See Robinson arithmetic and Recursive definition

## 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.

See Robinson arithmetic and Second-order 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.

See Robinson arithmetic and Set (mathematics)

## Set theory

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

See Robinson arithmetic and Set theory

## Set-theoretic definition of natural numbers

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

See Robinson arithmetic and Set-theoretic definition of 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.

See Robinson arithmetic and Skolem arithmetic

## Springer Science+Business Media

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.

See Robinson arithmetic and Springer Science+Business Media

## Successor function

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

See Robinson arithmetic and Successor function

## 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).

See Robinson arithmetic and Tennenbaum's theorem

## Total order

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

See Robinson arithmetic and Total order

## Unary operation

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

See Robinson arithmetic and Unary operation

## 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".

See Robinson arithmetic and Universal quantification

## Variable (mathematics)

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

See Robinson arithmetic and Variable (mathematics)

## 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).

See Robinson arithmetic and Zermelo set theory

## 0

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

## See also

### Formal theories of arithmetic

- Büchi arithmetic
- Bounded arithmetic
- Decidability of first-order theories of the real numbers
- Elementary function arithmetic
- Existential theory of the reals
- Heyting arithmetic
- Induction, bounding and least number principles
- Non-standard model of arithmetic
- Peano axioms
- Presburger arithmetic
- Primitive recursive arithmetic
- Robinson arithmetic
- Second-order arithmetic
- Skolem arithmetic
- Tarski's axiomatization of the reals
- True arithmetic
- Typographical Number Theory

## References

Also known as Q arithmetic, Robinson arithmetic Q, Robinson axioms, Robinson's Arithmetic, Robinson's Q.