# Finite Chain is Order-Isomorphic to Finite Ordinal

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## Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $C$ be a finite chain in $S$.

Then for some finite ordinal $\mathbf n$:

- $\struct {C, {\preceq \restriction_C} }$ is order-isomorphic to $\mathbf n$.

That is:

- $\struct {C, {\preceq \restriction_C} }$ is order-isomorphic to $\N_n$

where $\N_n$ is the initial segment of $\N$ determined by $n$:

- $\N_n = \set {k \in \N: k < n} = \set {0, 1, \ldots, n - 1}$

## Proof

By the definition of finite set:

- there exists an $n \in \N$ such that there exists a bijection $f: C \to \N_n$.

This $n$ is unique by Equality of Natural Numbers and Set Equivalence behaves like Equivalence Relation.

Define a mapping $g: \N_n \to C$ recursively as:

- $\map g 0 = \min C$
- $\map g {k + 1} = \map \min {C \setminus \map g {N_k} }$

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## Sources

- 1967: Garrett Birkhoff:
*Lattice Theory*(3rd ed.): $\S \text I.3$: Theorem $5$