I have an issue with a proof given "Théorie axiomatique des ensembles" by J.L Krivine.
Ordinal definition given of $\alpha$:
- $"x \in y"$ defines a strict, total, well-foundedness relation R in $\alpha$
- $x \in \alpha \Rightarrow x \subset \alpha$
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 $\xi = \emptyset$ ?