Existence of Minimal Uncountable Well-Ordered Set/Corollary 1

Corollary to Existence of Minimal Uncountable Well-Ordered Set

Let $\Omega$ denote the minimal uncountable well-ordered set.

The cardinality of $\Omega$ satisfies:

$\card \N < \card \Omega \le \mathfrak c$

where $\card \N$ is the cardinality of the natural numbers and $\mathfrak c$ is the cardinality of the continuum.

Proof

By the definition of $\Omega$ as a minimal uncountable well-ordered set:

$\card \N < \card \Omega$

by the definition of uncountable.

Furthermore:

$\card \Omega \le \card {\powerset \N}$ follows the from construction of $\Omega$ in the main proofs; $\Omega$ is a subset of $\powerset \N$.

This article, or a section of it, needs explaining. In particular: While a corollary is by definition something that follows directly on from another proof, house rules (somewhere) state that each page in $\mathsf{Pr} \infty \mathsf{fWiki}$ is to be self-contained and standalone. So at least a link is to be made to the specific result, or part of it, that contains "construction of $\Omega$". Optimum approach is to extract that construction and place it in its own page so it can then be linked to as appropriate from wherever it is required. I have not investigated that "construction of $\Omega$ in the main proof(s)", so whether it would work best as a lemma subpage or a result in its own right is something I can't immediately comment on. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

That $\card {\powerset \N} = \mathfrak c$ is shown in Power Set of Natural Numbers has Cardinality of Continuum.

Combining the above statements yields:

$\card \N < \card \Omega \le \mathfrak c$

$\blacksquare$

Sources

• 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications $\S P.18$
• 2000: James R. Munkres: Topology (2nd ed.) $\S 1.11$ Supplementary Exercise $8$