First uncountable ordinal

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In mathematics, the first uncountable ordinal, traditionally denoted by ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} or sometimes by Ω {\displaystyle \Omega } {\displaystyle \Omega }, is the smallest ordinal number that is the order type of an uncountable well-ordered set. It is the supremum (least upper bound) of all countable ordinals. In the von Neumann representation, the elements of ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} are the countable ordinals (including finite ordinals),[1] of which there are uncountably many.

The cardinality of the set ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} is the first uncountable cardinal number, ℵ 1 {\displaystyle \aleph _{1}} {\displaystyle \aleph _{1}} (aleph-one). The ordinal ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} is thus the initial ordinal of ⁠ ℵ 1 {\displaystyle \aleph _{1}} {\displaystyle \aleph _{1}}⁠. Like all other initial ordinals of infinite cardinals, ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} is a limit ordinal, i.e. there is no ordinal α {\displaystyle \alpha } {\displaystyle \alpha } such that ⁠ ω 1 = α + 1 {\displaystyle \omega _{1}=\alpha +1} {\displaystyle \omega _{1}=\alpha +1}⁠. Formally, cardinal numbers are usually represented as their initial ordinals, in which case ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} and ℵ 1 {\displaystyle \aleph _{1}} {\displaystyle \aleph _{1}} are considered equal as sets. More generally, for any ordinal ⁠ α {\displaystyle \alpha } {\displaystyle \alpha }⁠, ω α {\displaystyle \omega _{\alpha }} {\displaystyle \omega _{\alpha }} denotes the initial ordinal of the cardinal ⁠ ℵ α {\displaystyle \aleph _{\alpha }} {\displaystyle \aleph _{\alpha }}⁠.

The continuum hypothesis (CH) states that ⁠ ℶ 1 = ℵ 1 {\displaystyle \beth _{1}=\aleph _{1}} {\displaystyle \beth _{1}=\aleph _{1}}⁠ (where ⁠ ℶ 1 = 2 ℵ 0 = | R | {\displaystyle \beth _{1}=2^{\aleph _{0}}=\vert \mathbb {R} \vert } {\displaystyle \beth _{1}=2^{\aleph _{0}}=\vert \mathbb {R} \vert }⁠ is the second beth number), which implies that ⁠ | ω 1 | = | R | {\displaystyle \vert \omega _{1}\vert =\vert \mathbb {R} \vert } {\displaystyle \vert \omega _{1}\vert =\vert \mathbb {R} \vert }⁠, i.e., the countable ordinals are equinumerous to the real numbers. If CH does not hold, but the axiom of choice (AC) does, then ⁠ | ω 1 | {\displaystyle \vert \omega _{1}\vert } {\displaystyle \vert \omega _{1}\vert }⁠, as the smallest uncountable cardinal, is strictly less than ⁠ | R | {\displaystyle \vert \mathbb {R} \vert } {\displaystyle \vert \mathbb {R} \vert }⁠. If AC also does not hold then ⁠ | ω 1 | {\displaystyle \vert \omega _{1}\vert } {\displaystyle \vert \omega _{1}\vert }⁠ may be incomparable with ⁠ | R | {\displaystyle \vert \mathbb {R} \vert } {\displaystyle \vert \mathbb {R} \vert }⁠, but never larger than ⁠ | R | {\displaystyle \vert \mathbb {R} \vert } {\displaystyle \vert \mathbb {R} \vert }⁠.[2]

The existence of ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} does not depend on AC, as it can be constructed explicitly as the Hartogs number of ⁠ ω 0 = N {\displaystyle \omega _{0}=\mathbb {N} } {\displaystyle \omega _{0}=\mathbb {N} }⁠. More concretely, the set of all well-orderings on ⁠ N {\displaystyle \mathbb {N} } {\displaystyle \mathbb {N} }⁠ can be constructed as a subset of all binary relations on ⁠ N {\displaystyle \mathbb {N} } {\displaystyle \mathbb {N} }⁠, and thus applying the axiom of replacement to replace every well-ordering with its order type will give ⁠ ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}}⁠.

Topological properties

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} is often written as [ 0 , ω 1 ) {\displaystyle [0,\omega _{1})} {\displaystyle [0,\omega _{1})}, to emphasize that it is the space consisting of all ordinals smaller than ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}}.

If the axiom of countable choice holds, every increasing ω-sequence of elements of [ 0 , ω 1 ] {\displaystyle [0,\omega _{1}]} {\displaystyle [0,\omega _{1}]} converges to a limit in [ 0 , ω 1 ] {\displaystyle [0,\omega _{1}]} {\displaystyle [0,\omega _{1}]}. The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space [ 0 , ω 1 ) {\displaystyle [0,\omega _{1})} {\displaystyle [0,\omega _{1})} 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, [ 0 , ω 1 ) {\displaystyle [0,\omega _{1})} {\displaystyle [0,\omega _{1})} is first-countable, but neither separable nor second-countable.

The space [ 0 , ω 1 ] = ω 1 + 1 {\displaystyle [0,\omega _{1}]=\omega _{1}+1} {\displaystyle [0,\omega _{1}]=\omega _{1}+1} is compact and not first-countable. ω 1 {\displaystyle \omega _{1}} {\displaystyle \omega _{1}} is used to define the long line and the Tychonoff plank—two important counterexamples in topology.

See also

References

  1. "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-08-12.
  2. "first uncountable ordinal in nLab". ncatlab.org. Archived from the original on 2020-10-03. Retrieved 2020-08-12.

Bibliography