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:
- Is every transitive wellordered set an ordinal?
- 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?
- 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?