Existence of Minimal Uncountable Well-Ordered Set/Corollary 2

Corollary to Existence of Minimal Uncountable Well-Ordered Set

Let $X$ be a well-ordered set.

Then there exists a well-ordered set with cardinality strictly greater than $X$.

Proof

Follows from the same arguments proving the main result, mutatis mutandis, replacing:

"$\powerset \N$" with "$\powerset X$"

"Power Set of Natural Numbers is not Countable" with "No Bijection from Set to its Power Set"

"is uncountable" with "has cardinality $\size {\powerset X}$"

"is countable" with "has cardinality strictly less than $\size {\powerset X}$"

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) $\S 1.11$ Supplementary Exercise $8$