It is my understanding that ordinals can be defined in a variety of ways, the one I am looking at (this is in a context where ZF has not been formally introduced yet) is the following:
An ordinal is a transitive set that is well-ordered by $\in$
I think the aim of this definition is to avoid 'isomorphism classes' in a context where taking relations over classes have not been treated formally yet. Then, by the Mostowski collapse lemma, every well-ordered set $X$ is isomorphic to a unique ordinal.
But is the definition of a well-ordering not a total/linear order whose order is well-founded? In which case, transitivity is inherent to the definition? Is there a difference between a set being transitive and the relation on that set being transitive?
For instance, to prove every member of an ordinal is an ordinal, I found this:
let $\alpha$ be an ordinal and $\beta\in \alpha$. Since $\alpha$ is transitive, $\beta\subseteq \alpha$ and hence $\beta$ is linearly ordered by $\in$. To show $\beta$ is transitive, take $\delta\in \gamma\in \beta$, then both $\delta,\beta$ in $\in \alpha$ so either $\delta \in\beta$ or $\delta = \beta$ or $\beta\in \delta$, and only the first can hold otherwise we get a set with no least element.
This starts by proving $\beta$ is linearly ordered, so should it not automatically be transitive? I am missing a subtlety but I can't see what at the moment and would love some help!
Edit: made a correction pointed out in the comments
Best Answer
Every singleton is well-ordered. Only one singleton is transitive.
The key point is that a well-ordered is a transitive relation; but a transitive set is a set which is a subset of its own power set. These are separate technical terms.