In
mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the first uncountable ordinal, traditionally denoted by ω
1 or sometimes by Ω, is the smallest
ordinal number
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...
that, considered as a
set, is
uncountable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. It is the
supremum
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

(least upper bound) of all countable ordinals. The elements of ω
1 are the countable ordinals (including finite ordinals), of which there are uncountably many.
Like any ordinal number (in von Neumann's approach), ω
1 is a
well-ordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, with
set membership
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
("∈") serving as the order relation. ω
1 is a
limit ordinal
In set 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 an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ...
, i.e. there is no ordinal α with α + 1 = ω
1.
The
cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of the set ω
1 is the first uncountable
cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
, ℵ
1 (
aleph-one). The ordinal ω
1 is thus the
initial ordinal
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly ...
of ℵ
1. Under the
continuum hypothesis
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, the cardinality of ω
1 is the same as that of
—the set of
real numbers
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
.
In most constructions, ω
1 and ℵ
1 are considered equal as sets. To generalize: if α is an arbitrary ordinal, we define ω
α as the initial ordinal of the cardinal ℵ
α.
The existence of ω
1 can be proven without the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

. For more, see
Hartogs number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.
Topological properties
Any ordinal number can be turned into a
topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
by using the
order topology
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. When viewed as a topological space, ω
1 is often written as
1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_ω1.
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If_the_axiom_of_countable_choice_holds,_every_
1),_to_emphasize_that_it_is_the_space_consisting_of_all_ordinals_smaller_than_ω1.
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If_the_axiom_of_countable_choice_holds,_every_sequence">increasing_ω-sequence_of_elements_of_
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1)_converges_to_a_Limit_of_a_sequence">limit
Limit_or_Limits_may_refer_to:
_Arts_and_media
*_Limit_(music),_a_way_to_characterize_harmony
*_Limit_(song),_"Limit"_(song),_a_2016_single_by_Luna_Sea
*_Limits_(Paenda_song),_"Limits"_(Paenda_song),_2019_song_that_represented_Austria_in_the_Eurov_...
_in_[0,ω
1)._The_reason_is_that_the_union_(set_theory).html" ;"title=",ω
1)_converges_to_a_Limit_of_a_sequence.html" "title="sequence.html" ;"title="axiom_of_countable_choice.html" ;"title=",ω
1), to emphasize that it is the space consisting of all ordinals smaller than ω
1.
If the axiom of countable choice">,ω
1), to emphasize that it is the space consisting of all ordinals smaller than ω
1.
If the axiom of countable choice holds, every sequence">increasing ω-sequence of elements of [0,ω
1) converges to a Limit of a sequence">limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
in [0,ω