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I am just reading about Lawvere's ETCS axioms for set theory, mainly this paper: https://arxiv.org/pdf/1212.6543v1.pdf, where Tom Leinster introduces a simplified form of these axioms. I am wondering how could ordinal sets (and ordinal arithmetic, cardinal sets, cardinal arithmetic, etc.) be formalized within this theory. I mean, I know that you can define ordinals to be the isomorphism class of well-ordered sets, but, how could you give a specific characterization of the ordinals within Set? How can you properly study ordinal arithmetic within this theory?

The aim of this question is if an undergraduate course on Axiomatic Set Theory could be constructed that only relied on ETCS axioms (which I think are more related with the activity of the "working mathematician" than ZFC) and could cover the usual topics on ordinal and cardinal arithmetic and transfinite induction and recursion. That's why, although I know nowadays a lot of mathematicians and logicians prefer to understand sets within Topos Theory and HoTT, I think it would be interesting to formalize all these concepts on a simple axiomatic set theory as ETCS.

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  • $\begingroup$ I think you'd have a hard time dealing with better ordinals than "the isomorphism classes of wellorders on subsets of $X$" for particular $X$, though perhaps there's a workaround I'm forgetting. It might be a little awkward, but it could be done, with some care, mostly as normal (especially since the logic of ETCS is classical). $\endgroup$ – Malice Vidrine Jan 14 '18 at 0:20
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    $\begingroup$ Set is not really concerned with ordered types, as far as morphisms go... $\endgroup$ – Asaf Karagila Jan 14 '18 at 2:48
  • $\begingroup$ Note that in ZFC, "is an ordinal number" is not a property preserved by isomorphisms of sets. $\endgroup$ – Hurkyl Feb 4 '18 at 10:11
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You would "properly study ordinal arithmetic" by studying the arithmetic of well-ordered sets.

I'm guessing from your post you specifically have in mind the notion of transitively closed sets totally ordered by $\in$, but the reasons to think of such things don't really apply in this context, because:

  • we aren't very interested in choosing canonical representatives, and
  • we aren't studying $\in$-trees.

Doing arithmetic on well-ordered sets is even a fairly standard thing to do; for example, the first definitions of how to add and multiply ordinals at wikipedia's page on ordinal arithmetic are via constructions on well-ordered sets.

Similarly, you would "properly study cardinal arithmetic" by studying the arithmetic of sets. And again, wikipedia defines cardinal arithmetic via operations on arbitrary sets

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