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

  • $\begingroup$ Saying a set $A$ is transitive is short-hand for saying the relation $\in$ on the set is a transitive relation. $\endgroup$ – GEdgar Sep 23 '20 at 14:44
  • $\begingroup$ 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. $\endgroup$ – Andreas Blass Sep 23 '20 at 16:06
  • $\begingroup$ @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 $\endgroup$ – Sonk Sep 23 '20 at 16:22

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.

  • $\begingroup$ 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? $\endgroup$ – Sonk Sep 23 '20 at 16:16
  • $\begingroup$ That is correct. $\endgroup$ – Asaf Karagila Sep 23 '20 at 17:18
  • $\begingroup$ Thank you! Confusing wording $\endgroup$ – Sonk Sep 23 '20 at 17:20

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