# A normal filter including the tail sets is $\kappa$-complete

I need to show that

if $$\cal F$$ is a normal filter on a regular uncountable cardinal $$\kappa$$, and if $$\cal F$$ contains all tail sets, i.e. all $$C_\alpha=\{\beta \ : \ \alpha<\beta<\kappa\}$$ Then $$\cal F$$ is $$\kappa$$-complete

My thought

Let $$X$$ be a collection of less than $$\kappa$$ elements of $$\cal F$$, i.e. for some $$\lambda<\kappa$$: $$X = \{X_\alpha,\ \alpha<\lambda\}$$ With $$X_\alpha\in \cal F$$ for all $$\alpha$$.

We need to show that $$\bigcap_{\alpha<\lambda} X_\alpha \in \cal F$$. Let $$Y$$ be the collection $$X$$ adding the tail set with $$\alpha\geq\lambda$$: $$Y = \{X_\alpha:\ \alpha<\lambda\} \cup \{C_\alpha : \ \alpha \geq \lambda\}$$

Lets define $$Y_\alpha$$ as $$Y_\alpha = \left\{ \begin{array}{ll} X_\alpha&\text{ if } \alpha<\lambda\\ C_\alpha&\text{ if } \alpha\geq\lambda \end{array}\right.$$

For all $$\alpha$$, $$Y_\alpha\in\cal F$$, then because $$\cal F$$ is normal, $$\triangle_{\alpha<\kappa} Y_\alpha \in \cal F$$ But by definition of the diagonal intersection $$\triangle_{\alpha<\kappa} Y_\alpha = \{\beta<\kappa : \ \beta \in \bigcap_{\alpha<\beta}Y_\alpha\}$$ we have $$\triangle_{\alpha<\kappa}Y_\alpha \subseteq \bigcap_{\alpha<\lambda} X_\alpha$$. Therefore, because $$\cal F$$ is a filter $$\bigcap_{\alpha<\lambda} X_\alpha\in \cal F$$

My issue is with the statement $$\triangle_{\alpha<\kappa}Y_\alpha \subseteq \bigcap_{\alpha<\lambda} X_\alpha$$

Let $$\beta \in \triangle_{\alpha<\kappa}Y_\alpha = \{\beta<\kappa : \ \beta \in \bigcap_{\alpha<\beta}Y_\alpha\}$$

Clearly if $$\lambda < \beta < \kappa$$, then $$\beta\in\bigcap_{\alpha<\beta}Y_\alpha$$ and $$\beta\in\bigcap_{\alpha<\lambda}Y_\alpha=\bigcap_{\alpha<\lambda}X_\alpha$$

But if $$\beta < \lambda$$, how can I prove that $$\beta\in\bigcap_{\alpha<\lambda}X_\alpha$$ ?

• Intersect $X_\alpha$ with $C_\lambda$ in your definition of $Y_\alpha.$ That is still in $\mathcal F$ of course, and it takes care of the last part for you. – spaceisdarkgreen Mar 19 at 11:19