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In mathematics, the first uncountable ordinal, traditionally denoted by or sometimes by , is the smallest ordinal number that, considered as a set, is uncountable.It is the supremum (least upper bound) of all countable ordinals.
When λ = ω (the first infinite ordinal), the space [0,ω) is just N with the usual (still discrete) topology, while [0,ω] is the one-point compactification of N. Of particular interest is the case when λ = ω 1, the set of all countable ordinals, and the first uncountable ordinal.
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [ 1 ] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
Also, is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and is the order type of that set), is the smallest ordinal whose cardinality is greater than , and so on, and is the limit of for natural ...
For each Borel set B, there is some countable ordinal α B such that B can be obtained by iterating the operation over α B. However, as B varies over all Borel sets, α B will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω 1, the first uncountable ordinal.
More generally, the Zariski 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 \omega _{1}+1=\left[0,\omega _{1}\right]} where ω 1 {\displaystyle \omega _{1}} is the first uncountable ordinal number.
the first uncountable ordinal (also written as ω 1) [82] Chaitin's constant for a given computer program; the vacuum state in quantum field theory; represents: angular velocity / radian frequency (rad/sec) [83] the argument of periapsis in astronomy and orbital mechanics
Let stand for the first uncountable ordinal, or, in fact, any ordinal which is an -number and guaranteed to be greater than all the countable ordinals which will be constructed (for example, the Church–Kleene ordinal is adequate for our purposes; but we will work with because it allows the convenient use of the word countable in the definitions).