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How do you prove that every well-ordered set is isomorphic to a unique ordinal number?
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The proof for the stament that: "Every well-ordered set is isomorphic to a unique ordinal number."

Given a well-ordered set $W$, we find an isomorphic ordinal as follows: Define $F(x) = \alpha$ if $\alpha$ is isomorphic to the initial segment of $W$ given by $x$.

If such an $\alpha$ exists, then it is unique. By the Replacement Axioms, $F(W)$ is a set. For each $x \in W$, such an $\alpha$ exists (otherwise consider the least $x$ for which such an $\alpha$ does not exist). If $\gamma$ is the least $\gamma \not\in F(W)$ then $F(W) =\gamma$ and we have an isomorphism of $W$ onto $\gamma$.

by Diamond (75,934 points)

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