mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a setoid (''X'', ~) is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

(or

type Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * ...

) ''X'' equipped with an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

~. A setoid may also be called E-set,

Bishop A bishop is an ordained member of the clergy who is entrusted with a position of Episcopal polity, authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance and administration of di ...

set, or extensional set. Setoids are studied especially in

proof theory Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...

and in type-theoretic

foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...

. Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the

quotient set In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

(turning equivalence into

equality Equality generally refers to the fact of being equal, of having the same value. In specific contexts, equality may refer to: Society * Egalitarianism, a trend of thought that favors equality for all people ** Political egalitarianism, in which ...

). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of

intension In any of several fields of study that treat the use of signs—for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language—an intension is any property or quality connoted by a word, phrase, or another s ...

al equality (the equality on the original set) and extensional equality (the equivalence relation, or the equality on the quotient set).

Proof theory

In proof theory, particularly the proof theory of

constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...

based on the

Curry–Howard correspondence In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs. It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-p ...

, one often identifies a mathematical

proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

with its set of

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

s (if any). A given proposition may have many proofs, of course; according to the principle of proof irrelevance, normally only the truth of the proposition matters, not which proof was used. However, the Curry–Howard correspondence can turn proofs into

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

s, and differences between algorithms are often important. So proof theorists may prefer to identify a proposition with a ''setoid'' of proofs, considering proofs equivalent if they can be converted into one another through beta conversion or the like.

Type theory

In type-theoretic foundations of mathematics, setoids may be used in a type theory that lacks

quotient type In the field of type theory in computer science, a quotient type is a data type which respects a user-defined equality relation. A quotient type defines an equivalence relation \equiv on elements of the type - for example, we might say that two ...

s to model general mathematical sets. For example, in

Per Martin-Löf Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Sweden, Swedish logician, philosopher, and mathematical statistics, mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathe ...

intuitionistic type theory Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory (MLTT)) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematicia ...

, there is no type of

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, only a type of regular Cauchy sequences of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s. To do

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...

in Martin-Löf's framework, therefore, one must work with a ''setoid'' of real numbers, the type of regular Cauchy sequences equipped with the usual notion of equivalence. Predicates and functions of real numbers need to be defined for regular Cauchy sequences and proven to be compatible with the equivalence relation. Typically (although it depends on the type theory used), the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

will hold for functions between types (intensional functions), but not for functions between setoids (extensional functions). The term "set" is variously used either as a synonym of "type" or as a synonym of "setoid".

Constructive mathematics

, one often takes a setoid with an

apartness relation In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. An apartness relation is often written as \# (⧣ in unicode) to distinguish from the negation of equality ...

instead of an equivalence relation, called a constructive setoid. One sometimes also considers a partial setoid using a partial equivalence relation or partial apartness (see e.g. Barthe ''et al.'', section 1).

Notes

References

*. *.

External links

Implementation of setoids
in

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