Let $S$ be an infinite partially ordered set. Is there a name for the smallest ordinal number which is not ordinally similar to any subset of any chain of $S$? Assume that we are working in ZF. If the axiom of choice is needed to guarantee the existence of such an ordinal number, then assume that we are working in ZFC.
2026-03-30 02:05:34.1774836334
A question about well-ordered subsets of partially ordered sets
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Of course there is.
First note that by Hartogs theorem there is a least ordinal $\alpha$ which does not inject (regardless to the ordering) into $S$. Now consider the set: $$\{\beta\leq\alpha\mid\beta\text{ does not order embed into }S\}$$
The set is non-empty, since $\alpha$ does not order embed into $S$, since it is a non-empty set of ordinals, it has a least element.
No use of the axiom of choice was made here, and you can probably modify the proof of Hartogs theorem to obtain a direct proof of this.