From what I gather a set $x$ is $transitive$ if whenever $y \in z , z \in x \rightarrow y \in x$

And one of the properties of a well-ordered set is that $x \in y , y \in z \rightarrow x \in z$

Now I gather a set $x$ is an ordinal $\Leftrightarrow x$ is transitive and well-ordered by $\in$

My question is why is it necessary to say an ordinal is transitive, isn't it already covered in the definition of well-ordering?

(Sorry I'm probably missing something very obvious)


Take any singleton, $\{x\}$, then $\in$ well-orders $\{x\}$, since $\in\restriction(\{x\}\times\{x\})=\varnothing$ and $(\{x\}\,\varnothing)$ is a well-ordered set. However there is only one transitive singleton which is well-ordered by $\in$, which is $\{\varnothing\}$.

Another example can be $\{\omega,\omega_1\}$, which is also well-ordered by $\in$, since $\in\restriction(\{\omega,\omega_1\}\times\{\omega,\omega_1\})=\{(\omega,\omega_1)\}$ which is a well-ordering of $\{\omega,\omega_1\}$. And of course, any set of ordinals can be well-ordered using $\in$ that way. Regardless to being transitive or not.

The point is that $\in$ can be a transitive relation on a set, without the set being transitive.

  • $\begingroup$ Sorry could you define what $↾$ refers to? I am unfamiliar with the notation. Thanks. $\endgroup$ – mrhappysmile Jan 16 '15 at 15:27
  • 1
    $\begingroup$ It's the restriction of the $\in$ relation to the set which appears on the right of the $\restriction$ symbol. Namely, $R\restriction(A\times B)=R\cap(A\times B)=\{(a,b)\in A\times B\mid (a,b)\in R\}$. $\endgroup$ – Asaf Karagila Jan 16 '15 at 15:29
  • $\begingroup$ this makes sense - thank you! You explain everything so well, I appreciate it! $\endgroup$ – mrhappysmile Jan 16 '15 at 15:33
  • $\begingroup$ Thanks. I do my best... :-) $\endgroup$ – Asaf Karagila Jan 16 '15 at 15:38

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.