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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)

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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.

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  • $\begingroup$ Sorry could you define what $↾$ refers to? I am unfamiliar with the notation. Thanks. $\endgroup$ – mrhappysmile Jan 16 '15 at 15:27
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    $\begingroup$ 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

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