# Proof that the collection of ordinal set is well ordered from J.L Krivine book

I have an issue with a proof given "Théorie axiomatique des ensembles" by J.L Krivine.

Ordinal definition given of $$\alpha$$:

1. $$"x \in y"$$ defines a strict, total, well-foundedness relation R in $$\alpha$$
2. $$x \in \alpha \Rightarrow x \subset \alpha$$

We are trying to prove:

Given two ordinal sets $$\alpha$$ and $$\beta$$, one of these cases is true: $$\alpha \in \beta$$ or $$\beta \in \alpha$$ or $$\alpha = \beta$$

The proof starts by considering $$\xi = \alpha \cap \beta$$

The sketch of the proof is:

$$\xi \: initial \: segment \: of \: \alpha \Rightarrow \xi = \alpha \: or \: \xi \in \alpha$$

$$\xi \: initial \: segment \: of \: \beta \Rightarrow \xi = \beta \: or \: \xi \in \beta$$

4 cases are studied. The last case $$\xi \in \alpha$$ and $$\xi \in \beta$$ is excluded because it will lead to $$\xi \in \xi$$ (impossible for ordinals)

My question is: how do we know $$\xi \ne \emptyset$$ ? Why two random ordinal sets have a common element ? Or does the proof holds if $$\xi = \emptyset$$ ?

• Where does the proof uses that $\xi\neq\varnothing$? – Asaf Karagila Jun 13 at 8:12
• If $\xi = \emptyset$ then would it still be an initial segment of $\alpha$ ? – g.lahlou Jun 13 at 8:17
• Is the empty set a subset of $\alpha$? – Asaf Karagila Jun 13 at 8:17
• The empty set is a subset of $\alpha$ but it does not prove it is an element of $\alpha$, does it ? Every initial segment of $\alpha$ should be an element of $\alpha$, but is $\emptyset$ an element of every ordinal? – g.lahlou Jun 13 at 8:22
• Is $\varnothing$ a transitive set? Is $\in$ a well-order of it? – Asaf Karagila Jun 13 at 8:23

Okay, I think I may have an answer to my question. Thanks @Asaf for pointing me to the right direction. According to a comment in this thread, if an ordinal is not the emptyset, it contains the emptyset. So, if $$\xi$$ is the emptyset, it is still an initial segment of $$\alpha$$ and $$\beta$$ considering none of them is an emptyset. If $$\alpha$$ or $$\beta$$ is the emptyset, the theorem we are trying to prove is trivial.