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...

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 limit ordinal. 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 limit ordinal, 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 limit ordinal, The α-th infinite initial ordinal is written ωα. Its cardinality is written ℵα. Informally...

zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: ε 0 = ω ω ω ⋅ ⋅ ⋅ = sup { ω , ω ω...

contain a limit ordinal whenever they contain all sufficiently large ordinals below it. Any ordinal is, of course, an open subset of any larger ordinal. We...

complementation of sets maps Gm into itself for any limit ordinal m; moreover if m is an uncountable limit ordinal, Gm is closed under countable unions. For each...

weak limit cardinal, defined as the union of all the alephs before it; and in general ℵ λ {\displaystyle \aleph _{\lambda }} for any limit ordinal λ is...

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...

recursive ordinals. Since the successor of a recursive ordinal is recursive, the Church–Kleene ordinal is a limit ordinal. It is also the smallest ordinal that...

infinite ordinal α {\displaystyle \alpha } is a regular ordinal if it is a limit ordinal that is not the limit of a set of smaller ordinals that as a...

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...

{\Bigr \}},} where α {\displaystyle \alpha } is an ordinal and λ {\displaystyle \lambda } is a limit ordinal. The cardinal ℶ 0 = ℵ 0 {\displaystyle \beth _{0}=\aleph...

countable ordinal such that to every limit ordinal α < μ there is assigned a fundamental sequence (a strictly increasing sequence of ordinals whose supremum...

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories...

is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is...

2.  With an ordinal i as a subscript, denotes the ith limit ordinal that has a cardinality greater than that of all preceding ordinals. 3.  In computer...

an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose...

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...

are called limit cardinals; and by the above definition, if λ is a limit ordinal, then ℵ λ {\displaystyle \aleph _{\lambda }} is a limit cardinal. The...

P(\beta )} for all β < α {\displaystyle \beta <\alpha } ). Limit case: Prove that for any limit ordinal λ {\displaystyle \lambda } , P ( λ ) {\displaystyle P(\lambda...

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...

conditions: For every limit ordinal 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...

mentioned before is the special case where α is a finite ordinal or the first limit ordinal ω. This more generalized version extends the aforementioned...

considered by Mahlo were weakly Mahlo cardinals. If κ is a limit ordinal and the set of regular ordinals less than κ is stationary in κ, then κ is weakly Mahlo...

dancers Student club Women's club Youth club Club set, a subset of a limit ordinal in set theory Clubsuit, a family of combinatorial principles in set...

any ordinal number β, let Vβ+1 be the power set of Vβ: V β + 1 := P ( V β ) . {\displaystyle V_{\beta +1}:={\mathcal {P}}(V_{\beta }).} For any limit ordinal...