Cardinal

In set theory, the cardinal numbers (or just cardinals) are equivalence classes defined by the relation "there exists a bijection from set A onto set B". Whereas ordinal numbers may be thought of as "structures" of certain kinds of sets, cardinals are best described as the "sizes" of sets.

Ordinal

In set theory, an ordinal number, or simply ordinal, is an equivalence class of well-ordered sets under the relation of order isomorphism. Intuitively speaking, the ordinals form a number system that can be viewed as an extension of the natural numbers into infinite values. They are important to googology since they describe the growth rates of functions via the fast-growing hierarchy and other ordinal hierarchies, as well as their appearances in other meeting points between googology and set theory.

Limit Ordinal

A limit ordinal is a non-zero ordinal that doesn't have a predecessor. Formally, an ordinal α>0 is a limit ordinal iff ∄β:β+1=α. The least limit ordinal is ω, some of the next limit ordinals are ω×2 and ω×3, and limit ordinals can informally be thought of as "multiples of ω" due to how all limit ordinals are of the form ω×β . Some author allows 0 to be a limit ordinal and hence a limit ordinal is sometimes called a non-zero limit ordinal. The class of limit ordinals is often denoted by" " Lim.

Fundamental sequence

A fundamental sequence (FS for short) is an important concept in the study of ordinal hierarchies. If α is a limit ordinal with cofinality ω, a fundamental sequence for α is a monotonically increasing sequence of length ω consisting of ordinals, supremum of which is equal to α. Due to poor standardization in set theory, definitions of valid FS's vary. Some authors use "least strict upper bound" instead of "supremum," some relax the monotonicity condition to only require nondecreasing sequences, and some even allow fundamental sequences for successor ordinals.

Normal form

A normal form of an ordinal is a sort of an expression based on a predicate commonly written as =_"NF" depending on the context, which is used to uniquely express an ordinal by a fixed collection of constants and functions.

Transfinite induction

Transfinite induction is an extension of mathematical induction that is applicable to well-founded classes such as an ordinal notation, an ordinal, the class On of ordinals, and the class V of sets. It can be used in proving the statements about ordinals such as comparisons of fast-growing functions, and in constructing maps on On or V such as elementary operators of ordinals and the rank of sets.

Ordinal notation

An ordinal notation is, generally, a method for systematically naming ordinals (usually countable ones). More specifically, it is a (primitive) recursive well-ordering on a recursive set of finite strings in a finite alphabet, although in practice the term is applied to more general cases than this. The primitive recursiveness is often loosened to be recursiveness, because it is known that the notion of a notation equipped with a recursive well-ordering is essentially equivalent to that of an ordinal notation. The well-ordering induces an injective function onto ordinals below the limit given as its ordinal type, and hence it actually names all ordinals below the limit.