In mathematics, the first uncountable ordinal, traditionally denoted by ω 1 {\displaystyle \omega _{1}} or sometimes by Ω {\displaystyle \Omega } , 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...

below the first uncountable ordinal ω1; their supremum is called Church–Kleene ω1 or ωCK 1 (not to be confused with the first uncountable ordinal, ω1), described...

one means. Aleph number Beth number First uncountable ordinal Injective function Weisstein, Eric W. "Uncountably Infinite". mathworld.wolfram.com. Retrieved...

topology on an algebraic variety over an uncountable field is not first-countable. Another counterexample is the ordinal space ω 1 + 1 = [ 0 , ω 1 ] {\displaystyle...

countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω 1 {\displaystyle \omega _{1}} , the first uncountable ordinal. The...

produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The small Veblen ordinal θ Ω ω ( 0 ) {\displaystyle...

for any given ordinal notation there will be ordinals below ω1 (the first uncountable ordinal) that are not expressible. Such ordinals are known as large...

\omega _{1}} , the first uncountable ordinal, which is the set of all countable ordinals, analogously to how the Church-Kleene ordinal is the set of all...

naming all ordinals less than the Church–Kleene ordinal, which is a countable ordinal. Beyond the countable, the first uncountable ordinal is usually...

Ω = ω1 is the first uncountable ordinal. εΩ+1 is the first epsilon number after Ω = εΩ. ψ(α) is defined to be the smallest ordinal that cannot be constructed...

produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions. The large Veblen ordinal is sometimes denoted...

In mathematics, the Feferman–Schütte ordinal (Γ0) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such...

to encompass transfinite listings. Under this definition, the first uncountable ordinal ω 1 {\displaystyle \omega _{1}} can be enumerated by the identity...

behavior of functions. Chaitin's constant. In set theory, the first uncountable ordinal number, ω1 or Ω The absolute infinite proposed by Georg Cantor...

{\displaystyle \omega } is the first infinite ordinal and ω 1 {\displaystyle \omega _{1}} the first uncountable ordinal. The deleted Tychonoff plank is...

it is sequentially compact if and only if it is compact. The first uncountable ordinal with the order topology is an example of a sequentially compact...

{\displaystyle \Omega } stand for the first uncountable ordinal ω 1 {\displaystyle \omega _{1}} , or, in fact, any ordinal which is an ε {\displaystyle \varepsilon...

_{\omega _{1}}} , where ω 1 {\displaystyle \omega _{1}} is the first uncountable ordinal, so it could be either a successor cardinal or a limit cardinal...

counted with multiplicity the density parameter in cosmology the first uncountable ordinal (also written as ω1) Chaitin's constant for a given computer program...

mathematics, ψ0(Ωω), widely known as Buchholz's ordinal[citation needed], is a large countable ordinal that is used to measure the proof-theoretic strength...

not be unique. For example, let X be the set of ordinals at most equal to the first uncountable ordinal Ω, with the topology generated by "open intervals"...

{\displaystyle \triangle } being the symmetric difference operator). The first uncountable ordinal ω 1 {\displaystyle \omega _{1}} , equipped with its natural order...

club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor...

interest is the case when λ = ω1, the set of all countable ordinals, and the first uncountable ordinal. The element ω1 is a limit point of the subset [0,ω1)...

0 ) {\displaystyle \varphi (1,0,0,0)} , where Ω is the smallest uncountable ordinal. Ackermann's system of notation is weaker than the system introduced...

rationals. T is uncountable as it has a non-empty level Uα for each countable ordinal α which make up the first uncountable ordinal. This proves that...

The ordinal ε0 is still countable, as is any epsilon number whose index is countable. Uncountable ordinals also exist, along with uncountable epsilon...

since it is not locally finite. The space of ordinals at most equal to Ω, the first uncountable ordinal with the order topology is a compact topological...

(all ordinals α {\displaystyle \alpha } with cardinality | α | ≤ ℵ 0 {\displaystyle |\alpha |\leq \aleph _{0}} ), the first uncountable ordinal. Since...