In set theory, the ordinal numbers are objects that represent the order types of well-ordered sets. Is there a similar class of objects that represent the order types of well-founded sets? If so, are there any good resources on them?
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When you say 'order types of well-founded sets', what do you mean? A well-founded set need not be equipped with an order. – Clive Newstead Sep 16 '12 at 16:27
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@clive I take it to mean something like this: say that $S$ and $T$ are partially-ordered sets equipped with well-founded orders, and there exist $f$ and $g$ which are order-preserving maps from $S\to T$ and $T\to S$ respectively. Then say that $S$ and $T$ are "well-foundedly equivalent" and have the same "well-founded order type". Suppose that this is well-defined, and actually defines an equivalence relation on the class of well-founded partially-ordered sets. Then… – MJD Sep 16 '12 at 16:30
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Can't you think of the "order types" in question as a disjoint union of various ordinals representing disjoint chains in the partial order? – Elchanan Solomon Sep 16 '12 at 16:34
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The ordinals are extremely nice because they have a Cantor-Bernstein like theorem. If $\alpha\leq\beta$ and $\beta\leq\alpha$ then $\alpha=\beta$.
This is not true for general well-founded orders, as Brian's answer here shows. So for general well-founded orders embeddings do not make a partial order on isomorphism classes.
Generally speaking, though, if you just want to have representatives you can always just choose a representative from every isomorphism class, you can even make sure that these are sets and their relation is $\in$ relation.
The Mostowski collapse lemma says that if $(A,R)$ is well-founded and extensional then there exists a unique set $M$ such that $(A,R)\cong(M,\in)$, which begins to sound a bit more like what you are looking for. I suggest that you try and play with things from there.
Asaf Karagila
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