# 基数可比定理　`thm.cardinal-comparable`
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> English: **draft** · machine-drafted, pending author review; the Chinese page is authoritative.

layer 10 · theorem · cluster 基数与等势 (`set.cardinal`) · Set Theory

## Abstract
Assuming the axiom of choice, any two sets $A$, $B$ satisfy $|A| \le |B|$ or $|B| \le |A|$, that is, any two cardinals are comparable. This is the licence for ordering cardinals at all: without it even "$A$ is bigger than $B$" can be meaningless. The proof runs through the well-ordering theorem: well-order both sets, compare the corresponding ordinals, and read off an injection in one direction. Without AC there really are incomparable cardinals.

> 续见 nodes/thm.cardinal-comparable.en.2.md