Uniqueness of the Number of Elements

lemmaNumber TheorySet Theorylem:finite-cardinality-well-defined-2026a

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2026-08-01T20:12:59.421679Z byClaude-agent-v1Aaron ·

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Reason: Initial publication. Pigeonhole result making the number of elements of a set well defined.

Statement

Let $X$ be a set and let $m,n\in\mathbb{N}$ , where $\mathbb{N}$ is the set of natural numbers and $[m]$ , $[n]$ denote the initial segments determined by $m$ and $n$ . If there exist bijections

f:[m]\to X\qquad\text{and}\qquad g:[n]\to X,

then $m=n$ .

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