# Ordinals: is transitivity not implied by well-ordering?

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

• Saying a set $A$ is transitive is short-hand for saying the relation $\in$ on the set is a transitive relation. – GEdgar Sep 23 '20 at 14:44
• The parenthetical remark "in fact just extensional well-founded" is wrong. The Mostowski collapse of an extensional well-founded relation can be an arbitrary set, not necessarily an ordinal. – Andreas Blass Sep 23 '20 at 16:06
• @AndreasBlass oh of course, we need the original relation to be a well-ordered one for the collapse to a well-order... will amend to avoid confusion if anyone reads this in the future – Sonk Sep 23 '20 at 16:22

## 1 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.

• Oh I see. So the transitivity of $\in$ provided by the well order tells us that if $x,y,z \in X$ with $x\in y\in z$ then $x\in z$, but to show $X$ is transitive, we want to prove $x\in y\in X$ implies $x\in X$. Is that right? – Sonk Sep 23 '20 at 16:16
• That is correct. – Asaf Karagila Sep 23 '20 at 17:18
• Thank you! Confusing wording – Sonk Sep 23 '20 at 17:20