Grothendieck universeIn mathematics, a Grothendieck universe is a set U with the following properties: If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.) If x and y are both elements of U, then is an element of U. If x is an element of U, then P(x), the power set of x, is also an element of U. If is a family of elements of U, and if I is an element of U, then the union is an element of U. A Grothendieck universe is meant to provide a set in which all of mathematics can be performed.
Ordinal numberIn set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element").
Universe (mathematics)In mathematics, and particularly in set theory, , type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem. These classes can serve as inner models for various axiomatic systems such as ZFC or Morse–Kelley set theory. Universes are of critical importance to formalizing concepts in inside set-theoretical foundations.
Universal setIn set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.
Constructible universeIn mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".
Axiom schema of replacementIn set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set.
Well-founded relationIn mathematics, a binary relation R is called well-founded (or wellfounded or foundational) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is, an element m ∈ S not related by s R m (for instance, "s is not smaller than m") for any s ∈ S. In other words, a relation is well founded if Some authors include an extra condition that R is set-like, i.e., that the elements less than any given element form a set.
Cumulative hierarchyIn mathematics, specifically set theory, a cumulative hierarchy is a family of sets indexed by ordinals such that If is a limit ordinal, then Some authors additionally require that or that . The union of the sets of a cumulative hierarchy is often used as a model of set theory. The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy of the von Neumann universe with introduced by . A cumulative hierarchy satisfies a form of the reflection principle: any formula in the language of set theory that holds in the union of the hierarchy also holds in some stages .
Reflection principleIn set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that, with respect to any given property, resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble". Weak forms of the reflection principle are theorems of ZF set theory due to , while stronger forms can be new and very powerful axioms for set theory. The name "reflection principle" comes from the fact that properties of the universe of all sets are "reflected" down to a smaller set.
Axiom of limitation of sizeIn set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is a member of a class is a set; a class that is not a set is a proper class.
