Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω. Limit ordinals are those that are non-zero and have no predecessor, such as ω or ω2
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal.
For example, the smallest limit ordinal is ω, the smallest ordinal greater than every natural number. This is a limit ordinal because for any smaller ordinal (i.e., for any natural number) n we can find another natural number larger than it (e.g. n+1), but still less than ω. The next-smallest limit ordinal is ω+ω. This will be discussed further in the article.
Using the von Neumann definition of ordinals, every ordinal is the well-ordered set of all smaller ordinals. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal. Using von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite...
or a limitordinal. Using von Neumann's ordinal numbers (the standard model of the ordinals used in set theory), the successor S(α) of an ordinal number...
the order relation. ω 1 {\displaystyle \omega _{1}} is a limitordinal, i.e. there is no ordinal α {\displaystyle \alpha } such that ω 1 = α + 1 {\displaystyle...
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...
ℵα+1 = (ℵα)+ ℵλ = ⋃{ ℵα | α < λ } for λ an infinite limitordinal, The α-th infinite initial ordinal is written ωα. Its cardinality is written ℵα. Informally...
contain a limitordinal whenever they contain all sufficiently large ordinals below it. Any ordinal is, of course, an open subset of any larger ordinal. We...
smallest infinite ordinal. The least such ordinal is ε0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite...
complementation of sets maps Gm into itself for any limitordinal m; moreover if m is an uncountable limitordinal, Gm is closed under countable unions. For each...
countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance...
weak limit cardinal, defined as the union of all the alephs before it; and in general ℵ λ {\displaystyle \aleph _{\lambda }} for any limitordinal λ is...
recursive ordinals. Since the successor of a recursive ordinal is recursive, the Church–Kleene ordinal is a limitordinal. It is also the smallest ordinal that...
functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908). If φ0 is any normal function, then for any non-zero ordinal α, φα is the...
singular ordinal is any ordinal that is not regular. Every regular ordinal is the initial ordinal of a cardinal. Any limit of regular ordinals is a limit of...
infinite ordinal α {\displaystyle \alpha } is a regular ordinal if it is a limitordinal that is not the limit of a set of smaller ordinals that as a...
2. With an ordinal i as a subscript, denotes the ith limitordinal that has a cardinality greater than that of all preceding ordinals. 3. In computer...
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories...
P(\beta )} for all β < α {\displaystyle \beta <\alpha } ). Limit case: Prove that for any limitordinal λ {\displaystyle \lambda } , P ( λ ) {\displaystyle P(\lambda...
<\lambda \},} where α {\displaystyle \alpha } is an ordinal and λ {\displaystyle \lambda } is a limitordinal. The cardinal ℶ 0 = ℵ 0 {\displaystyle \beth _{0}=\aleph...
mentioned before is the special case where α is a finite ordinal or the first limitordinal ω. This more generalized version extends the aforementioned...
Church–Kleene ordinal, the first nonrecursive ordinal, and denoted by ω 1 C K {\displaystyle \omega _{1}^{\mathsf {CK}}} . The Church–Kleene ordinal is a limit ordinal...
dancers Student club Women's club Youth club Club set, a subset of a limitordinal in set theory Clubsuit, a family of combinatorial principles in set...
conditions: For every limitordinal γ (i.e. γ is neither zero nor a successor), it is the case that f (γ) = sup{f (ν) : ν < γ}. For all ordinals α < β, it is the...
inaccessible. An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and...
particular Γ0 is the Feferman–Schütte ordinal. δ 1. A delta number is an ordinal of the form ωωα 2. A limitordinal Δ (Greek capital delta, not to be confused...
are called limit cardinals; and by the above definition, if λ is a limitordinal, then ℵ λ {\displaystyle \aleph _{\lambda }} is a limit cardinal. The...
considered by Mahlo were weakly Mahlo cardinals. If κ is a limitordinal and the set of regular ordinals less than κ is stationary in κ, then κ is weakly Mahlo...
Global Information