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# Robinson arithmetic

In mathematics, Robinson arithmetic, or Q, is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out in R. M. Robinson (1950). 

57 relations: Addition, Alfred Tarski, Andrzej Mostowski, Axiom, Axiom of adjunction, Axiom schema, Binary operation, Cardinality, Complete theory, Computable function, Concatenation, Converse (logic), Domain of a function, Elliott Mendelson, Empty set, Equality (mathematics), Existential quantification, Extensionality, First-order logic, Gödel numbering, Gödel's incompleteness theorems, General set theory, Gentzen's consistency proof, George Boolos, Infinite set, Injective function, Interpretation (logic), John Lucas (philosopher), John P. Burgess, Joseph R. Shoenfield, List of first-order theories, Mathematical induction, Mathematics, Multiplication, Natural number, Non-standard model of arithmetic, Operation (mathematics), Peano axioms, Polish notation, Raphael M. Robinson, Raymond Smullyan, Recursive definition, Richard Jeffrey, Second-order arithmetic, Set (mathematics), Set theory, Set-theoretic definition of natural numbers, Springer Science+Business Media, Successor function, Tennenbaum's theorem, ... Expand index (7 more) »

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

## Alfred Tarski

Alfred Tarski (January 14, 1901 &ndash; October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews,, School of Mathematics and Statistics, University of St Andrews.

## Andrzej Mostowski

Andrzej Mostowski (1 November 1913 – 22 August 1975) was a Polish mathematician.

## Axiom

An axiom or postulate 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 on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.

## 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, for every formula in the theory's language, that formula or its negation is demonstrable.

## Computable function

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

## Concatenation

In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end.

## Converse (logic)

In logic, the converse of a categorical or implicational statement is the result of reversing its two parts.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Elliott Mendelson

Elliott Mendelson (born 1931) is an American logician.

## Empty set

In mathematics, and more specifically set theory, the empty set or null 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.

## First-order logic

First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

## 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 demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.

## 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.

## George Boolos

George Stephen Boolos (September 4, 1940 – May 27, 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.

## Infinite set

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

## Injective function

In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.

## Interpretation (logic)

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

## John Lucas (philosopher)

John Randolph Lucas FBA (born 18 June 1929) is a British philosopher.

## John P. Burgess

John Patton Burgess (born 5 June 1948) is a John N. Woodhull Professor of Philosophy at Princeton University.

## Joseph R. Shoenfield

Joseph Robert Shoenfield (1927, Detroit – November 15, 2000, Durham, North Carolina) was an American mathematical logician.

## List of first-order theories

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

## Mathematical induction

Mathematical induction is a mathematical proof technique.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

## Natural number

In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").

## 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.

## Operation (mathematics)

In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value.

## 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.

## 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 reverse Polish notation (RPN) in which operators follow their operands.

## Raphael M. Robinson

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

## Raymond Smullyan

Raymond Merrill Smullyan (May 25, 1919 – February 6, 2017) was an American mathematician, magician, concert pianist, logician, Taoist, and philosopher.

## Recursive definition

A recursive definition (or inductive definition) in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set (Aczel 1978:740ff).

## Richard Jeffrey

Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist.

## Second-order arithmetic

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

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Set-theoretic definition of natural numbers

Several ways have been proposed to construct the natural numbers using set theory.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes 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 is a primitive recursive function S such that S(n).

## 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 linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

## Unary operation

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

## Universal quantification

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

## Variable (mathematics)

In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.

## Wolfgang Rautenberg

Wolfgang Rautenberg (27 February 1936 − 4 September 2011) was a German mathematician and logician whose areas of research were model theory, non-classical logic, modal logic, temporal logic and self reference.

## Zermelo set theory

Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

## References

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