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Definition. An ordered set is wellordered if each of its nonempty subsets has a least element.

Some definitions say an ordinal is an equivalence class of order-isomorphisms of wellordered sets, which I think is because these isomorphisms have nice properties (the only order-endomorphism on a wellordered set is the identity (automorphism), and given any two distinct wellordered sets one of them is isomorphic to an initial segment of the other).

But this definition doesn't leave me too happy, since we're taking an equivalence relation on the (proper?) class of all wellordered sets. How are we even guaranteed that every single equivalence class in this relation is a set?

Also, when trying to 'visualize' what an ordinal is, it's not too helpful to think of them as massive blobs of wellordered sets, since wellordered sets seem very unwieldy to me in general, particularly if their cardinality is large.

So I ask a bunch of questions:

  1. Is every transitive wellordered set an ordinal?
  2. We say $\phi:A\rightarrow B$ is an isomorphism of wellordered sets $A$ and $B$ if it preserves order and its inverse also does, where 'preserving order' means that $a_1<_Aa_2$ implies $\phi(a_1)<_B\phi(a_2)$ for any $a_1,a_2\in A$. What are some equivalent conditions to two wellordered sets being isomorphic? Is there even enough information in the definition to actually find equivalent conditions?
  3. I know every finite totally ordered set is wellordered, which makes me think: is every finite wellordered set an ordinal? (This would make every finite totally ordered set an ordinal.) Moreover, are the finite ordinals precisely the finite wellordered sets?

Now, in my mind, the natural numbers of everyday mathematics and the finite ordinals are not necessarily the same thing, but, for some reason, they seem to have exactly the same properties so that we can 'identify' the natural numbers of everyday mathematics with the finite ordinals of ZFC. So, leaving all rigor aside, I ask naively: what is the relationship between the natural numbers of everyday mathematics and the finite ordinals of ZFC? Are the former simply a 'model' of the abstract notion of an ordinal? Can we construct different 'models' of the natural numbers and a different arithmetics that don't match the behavior of the finite ordinals of ZFC? (I think of the ordinals are an 'abstract system' satisfying certain first-order-logic properties.)

This leads me to think, what is ZFC, precisely (or any collection of axioms, for that matter)? Is it a 'blueprint' for a mathematical universe from which we can construct concrete instances (what I hear some people call 'models')?

What are the natural numbers of everyday mathematics, anyway? Are they a 'model' of the natural numbers of Peano arithmetic? In which 'mathematical universe' do we do mathematics (when we compute integrals, and prove theorems on convergence of mappings, and group representation, and the such? Is it in an 'abstract system' like ZFC, or does all mathematics implicitly happen in a concrete 'model' of ZFC?

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    $\begingroup$ There's too many questions in this question. But, you might like the Von Neumann definition of ordinals, which bypasses your concern about the "equivalence class" definition. en.wikipedia.org/wiki/Ordinal_number $\endgroup$ – Lee Mosher Jan 16 '16 at 18:38
  • $\begingroup$ Yeah I was kinda rambling on about these other questions hoping someone might also answer them, but I guess that didn't happen! $\endgroup$ – étale-cohomology Jan 17 '16 at 10:28
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According to the usual (Von Neumann) definition, a transitive set is an ordinal if and only if it is well-ordered by the membership relation $\in$.

There are plenty of transitive sets that are not ordinals, though they can be well-ordered by a different relation than $\in$. A concrete example would be $V_\omega$, the set of all hereditarily finite sets. (Or, in general, the transitive closure of any set that is not itself an ordinal).

The way this connects with the "unwieldy" concept of equivalence classes of well-orders is that every well-ordered set is order-isomorphic to exactly one ordinal -- this can be proved by induction among the initial segments of the original well-ordered set. Thus the Von Neumann ordinals are useful representatives for the equivalence classes.

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No. Every singleton is well-ordered, but the only transitive singleton is $\{\varnothing\}$. Well, at least under the axiom of regularity.

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