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Bourbaki also defines an inductive set to be a partially ordered set that satisfies the hypothesis of Zorn's lemma when nonempty.. In descriptive set theory, an inductive set of real numbers (or more generally, an inductive subset of a Polish space) is one that can be defined as the least fixed point of a monotone operation definable by a positive Σ 1 n formula, for some natural number n ...
HoTT differs from ITT by its identity type (equality). Higher inductive types not only define a new type with constants and functions that create elements of the type, but also new instances of the identity type that relate them. A simple example is the circle type, which is defined with two constructors, a basepoint; base : circle. and a loop;
Recursive data structures can dynamically grow to an arbitrarily large size in response to runtime requirements; in contrast, a static array's size requirements must be set at compile time. Sometimes the term "inductive data type" is used for algebraic data types which are not necessarily recursive.
Most recursive definitions have two foundations: a base case (basis) and an inductive clause. The difference between a circular definition and a recursive definition is that a recursive definition must always have base cases, cases that satisfy the definition without being defined in terms of the definition itself, and that all other instances in the inductive clauses must be "smaller" in some ...
A set is called inductively defined if for some monotonic operator : (), () =, where () denotes the least fixed point of .The language of ID 1, , is obtained from that of first-order number theory, , by the addition of a set (or predicate) constant I A for every X-positive formula A(X, x) in L N [X] that only contains X (a new set variable) and x (a number variable) as free variables.
Inductive data type may refer to: Algebraic data type, a datatype each of whose values is data from other datatypes wrapped in one of the constructors of the datatype; Inductive family, a family of inductive data types indexed by another type or value; Recursive data type, a data type for values that may contain other values of the same type
In set theory, -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction. The principle implies transfinite induction and recursion.
Inductive programming (IP) is a special area of automatic programming, covering research from artificial intelligence and programming, which addresses learning of typically declarative (logic or functional) and often recursive programs from incomplete specifications, such as input/output examples or constraints.