successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor 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...

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...

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...

program Successor (EP), an EP by Sonata Arctica Successor (album), an album by Dedekind Cut A successor cardinal A successor ordinal The successor function...

cardinal. If λ were a successor ordinal, then ℵλ would be a successor cardinal and hence not weakly inaccessible. If λ were a limit ordinal less than κ then...

define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides...

spaces, and no ordinal beyond that is discrete. The ordinal α is compact as a topological space if and only if α is either a successor ordinal or zero. The...

the characterization of computability by recursive functions. Successor ordinal Successor cardinal Increment and decrement operators Sequence Steffen,...

based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor...

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...

} if λ is a successor ordinal then ℵ λ {\displaystyle \aleph _{\lambda }} is not a weak limit. Conversely, if a cardinal κ is a successor cardinal, say...

Regnal numbers are ordinal numbers used to distinguish among persons with the same name who held the same office. Most importantly, they are used to distinguish...

{\Bigr \}}} for any limit ordinal λ {\displaystyle \lambda } Note that this behavior is different from that of successor ordinals. Cardinalities less than...

finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal. If X and Y are disjoint...

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...

system of ordinal notations. Briefly, an ordinal notation is either the name zero (describing the ordinal 0), or the successor of an ordinal notation (describing...

ordinal β 1.  βX is the Stone–Čech compactification of X 2.  An ordinal γ A gamma number, an ordinal of the form ωα Γ The Gamma function of ordinals....

used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term...

successor of a computable ordinal is computable, and the set of all computable ordinals is closed downwards. The supremum of all computable ordinals is...

cofinality of any successor ordinal is 1. The cofinality of any nonzero limit ordinal is an infinite regular cardinal. A regular ordinal is an ordinal that is equal...

continuous: for every limit ordinal λ {\displaystyle \lambda } (i.e. λ {\displaystyle \lambda } is neither zero nor a successor), f ( λ ) = sup { f ( α )...

rank strictly less than α, regardless of whether α is 0, a successor ordinal, or a limit ordinal: V α := ⋃ β < α P ( V β ) . {\displaystyle V_{\alpha }:=\bigcup...

In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members...

transitive set x is an ordinal, x is a limit ordinal, x is a successor ordinal x is a finite ordinal The first infinite ordinal ω x is an ordered pair...

information Multiset Musical set theory Ordinal number Infinite descending chain Limit ordinal Successor ordinal Transfinite induction ∈-induction Well-founded...

ordinal α. The Morley rank is at least 0 if S is non-empty. For α a successor ordinal, the Morley rank is at least α if in some elementary extension N of...

regarded as ordinal notations. It contains ordinal notations for every computable ordinal, that is, ordinals below Church–Kleene ordinal, ω 1 CK {\displaystyle...

{\displaystyle P(0)} is true. Successor case: Prove that for any successor ordinal α + 1 {\displaystyle \alpha +1} , P ( α + 1 ) {\displaystyle P(\alpha...

S (corresponding to the limit ordinal case) or by appending b(S) to the "last" S (corresponding to the successor ordinal case). This proof shows that actually...