First uncountable ordinal

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the first uncountable ordinal, traditionally denoted by 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 or sometimes by 
 
 
 
 Ω
 
 
 {\displaystyle \Omega }
 
, is the smallest <a href="/facts/Ordinal_number/GelFLjJt">ordinal number</a> that, considered as a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>, is <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a>. It is the <a href="/facts/Supremum/oXCKVopz">supremum</a> (least upper bound) of all countable ordinals. When considered as a set, the elements of 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 are the countable ordinals (including finite ordinals), of which there are uncountably many.
Like any ordinal number (in <a href="/facts/Von_Neumann_ordinal/GelFLjJt">von Neumann's approach</a>), 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 is a <a href="/facts/Well-order/GrWEBWlW">well-ordered set</a>, with <a href="/facts/Set_membership/Q2mx1vHL">set membership</a> serving as the order relation. 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 is a <a href="/facts/Limit_ordinal/jPCCoPuP">limit ordinal</a>, i.e. there is no ordinal 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 such that 
 
 
 
 
 ω
 
 1
 
 
 =
 α
 +
 1
 
 
 {\displaystyle \omega _{1}=\alpha +1}
 
.
The <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of the set 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 is the first uncountable <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a>, 
 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \aleph _{1}}
 
 (<a href="/facts/Aleph-one/kwe9I4v5">aleph-one</a>). The ordinal 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 is thus the <a href="/facts/Ordinal_number/GelFLjJt">initial ordinal</a> of 
 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \aleph _{1}}
 
. Under the <a href="/facts/Continuum_hypothesis/R0yGqHND">continuum hypothesis</a>, the cardinality of 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 is 
 
 
 
 
 ℶ
 
 1
 
 
 
 
 {\displaystyle \beth _{1}}
 
, the same as that of 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
—the set of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>.
In most constructions, 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 and 
 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \aleph _{1}}
 
 are considered equal as sets. To generalize: if 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is an arbitrary ordinal, we define 
 
 
 
 
 ω
 
 α
 
 
 
 
 {\displaystyle \omega _{\alpha }}
 
 as the initial ordinal of the cardinal 
 
 
 
 
 ℵ
 
 α
 
 
 
 
 {\displaystyle \aleph _{\alpha }}
 
.
The existence of 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 can be proven without the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>. For more, see <a href="/facts/Hartogs_number/pLNz06f9">Hartogs number</a>.

First uncountable ordinal open-in-new

First uncountable ordinal