Setoid

Last updated

In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~. A setoid may also be called E-set, Bishop set, or extensional set. [1]

Contents

Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set (turning equivalence into equality). In contrast, setoids may be used when a difference between identity and equivalence must be maintained, often with an interpretation of intensional 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 based on the Curry–Howard correspondence, one often identifies a mathematical proposition with its set of proofs (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 algorithms, 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 types to model general mathematical sets. For example, in Per Martin-Löf's intuitionistic type theory, there is no type of real numbers, only a type of regular Cauchy sequences of rational numbers. To do real analysis 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 will hold for functions between types (intensional functions), but not for functions between setoids (extensional functions).[ clarification needed ] The term "set" is variously used either as a synonym of "type" or as a synonym of "setoid". [2]

Constructive mathematics

In constructive mathematics, one often takes a setoid with an apartness relation 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).

See also

Notes

  1. ↑ Alexandre Buisse, Peter Dybjer, "The Interpretation of Intuitionistic Type Theory in Locally Cartesian Closed Categories—an Intuitionistic Perspective", Electronic Notes in Theoretical Computer Science 218 (2008) 21–32.
  2. ↑ "Bishop's set theory" (PDF): 9.{{cite journal}}: Cite journal requires |journal= (help)

Related Research Articles

<span class="mw-page-title-main">Axiom of choice</span> Axiom of set theory

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

In the philosophy of mathematics, constructivism asserts that it is necessary to find a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.

In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and type theory in general is the academic study of type systems.

<span class="mw-page-title-main">Cantor's diagonal argument</span> Proof in set theory

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values.

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same.

Intuitionistic type theory is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions. However, all versions keep the core design of constructive logic using dependent types.

<span class="mw-page-title-main">Per Martin-Löf</span> Swedish logician, philosopher, and mathematical statistician

Per Erik Rutger Martin-Löf is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin-Löf's publications have been mainly in logic. In philosophical logic, Martin-Löf has wrestled with the philosophy of logical consequence and judgment, partly inspired by the work of Brentano, Frege, and Husserl. In mathematical logic, Martin-Löf has been active in developing intuitionistic type theory as a constructive foundation of mathematics; Martin-Löf's work on type theory has influenced computer science.

In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, and Idris, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.

In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it.

In mathematics, a partial equivalence relation is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, then the relation is an equivalence relation.

Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.

Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.

In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as to distinguish from the negation of equality which is weaker.

In mathematics, the effective topos introduced by Martin Hyland (1982) captures the mathematical idea of effectivity within the category theoretical framework.

<span class="mw-page-title-main">Homotopy type theory</span> Type theory in logic and mathematics

In mathematical logic and computer science, homotopy type theory refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies.

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 on elements of the type - for example, we might say that two values of the type Person are equivalent if they have the same name; formally p1 == p2 if p1.name == p2.name. In type theories which allow quotient types, an additional requirement is made that all operations must respect the equivalence between elements. For example, if f is a function on values of type Person, it must be the case that for two Persons p1 and p2, if p1 == p2 then f(p1) == f(p2).

Univalent foundations are an approach to the foundations of mathematics in which mathematical structures are built out of objects called types. Types in univalent foundations do not correspond exactly to anything in set-theoretic foundations, but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points of a space connected by a path. Univalent foundations are inspired both by the old Platonic ideas of Hermann Grassmann and Georg Cantor and by "categorical" mathematics in the style of Alexander Grothendieck. Univalent foundations depart from the use of classical predicate logic as the underlying formal deduction system, replacing it, at the moment, with a version of Martin-Löf type theory. The development of univalent foundations is closely related to the development of homotopy type theory.

References

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.