<mathematics> An axiom schema of set theory which states: if P(x) is a property then x : P is a set. I.e. all the things with some property form a set. Acceptance of this axiom leads to Russell's Paradox which is why Zermelo set theory replaces it with a restricted form. (1995-03-31)

In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements are the same set.

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads:

A list comprehension is a syntactic construct available in some programming languages for creating a list based on existing lists. It follows the form of the mathematical set-builder notation (set comprehension) as distinct from the use of map and filter functions.

<functional programming> An expression in a {functional language} denoting the results of some operation on (selected) elements of one or more lists. An example in Haskell: [ (x,y) | x <- [1 .. 6], y <- [1 .. x], x+y < 10] This returns all pairs of numbers (x,y) where x and y are elements of the list 1, 2, ..., 10, y <= x and their sum is less than 10. A list comprehension is simply "syntactic sugar" for a combination of applications of the functions, concat, map and filter. For instance the above example could be written: filter p (concat (map ( x -> map ( y -> (x,y)) [1..x]) [1..6])) where p (x,y) = x+y < 10 According to a note by Rishiyur Nikhil <nikhil@crl.dec.com>, (August 1992), the term itself seems to have been coined by Phil Wadler circa 1983-5, although the programming construct itself goes back much further (most likely Jack Schwartz and the SETL language). The term "list comprehension" appears in the references below. The earliest reference to the notation is in Rod Burstall and John Darlington's description of their language, NPL. David Turner subsequently adopted this notation in his languages SASL, KRC and Miranda, where he has called them "{ZF expressions}", set abstractions and list abstractions (in his 1985 FPCA paper [Miranda: A Non-Strict Functional Language with Polymorphic Types]). ["The OL Manual" Philip Wadler, Quentin Miller and Martin Raskovsky, probably 1983-1985]. ["How to Replace Failure by a List of Successes" FPCA September 1985, Nancy, France, pp. 113-146]. (1995-02-22)

<programming> (After Zermelo Frankel set theory). {David Turner}'s name for list comprehension. (1995-03-27)

In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary sets.

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