0
$\begingroup$

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

$\endgroup$
3
  • $\begingroup$ 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
3
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.