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Concept# First uncountable ordinal

Summary

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 considered as a set, the elements of are the countable ordinals (including finite ordinals), of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), is a well-ordered set, with set membership serving as the order relation. is a limit ordinal, i.e. there is no ordinal such that .
The cardinality of the set is the first uncountable cardinal number, (aleph-one). The ordinal is thus the initial ordinal of . Under the continuum hypothesis, the cardinality of is , the same as that of —the set of real numbers.
In most constructions, and are considered equal as sets. To generalize: if is an arbitrary ordinal, we define as the initial ordinal of the cardinal .
The existence of can be proven without the axiom of choice. For more, see Hartogs number.
Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, is often written as , to emphasize that it is the space consisting of all ordinals smaller than .
If the axiom of countable choice holds, every increasing ω-sequence of elements of converges to a limit in . The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.
The topological space is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, is first-countable, but neither separable nor second-countable.
The space is compact and not first-countable. is used to define the long line and the Tychonoff plank—two important counterexamples in topology.
Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, .

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Ontological neighbourhood

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In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0.

Ordinal number

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element").

Hartogs number

In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X.

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