Strong Finite Intersection Property and Pseudo-intersection (Kunen III.1.23)

Solution 1:

We need only finite subsets (from $J$) to "force" the pseudo-intersection of $\mathcal{F}^\ast$ to have the right property and to have this family has the SFIP in the first place.

I'll give the proof that $I$ is infinite in even more detail, I think I see a subtle use of finiteness there:

Let $k \in \omega$ arbitrary (a pregiven lower bound introduced by me, unrelated to $l$). The sets $Z_1 \cap \ldots Z_n \cap H_1$, $Z_1 \cap \ldots Z_n \cap H_2$ up to $Z_1 \cap \ldots Z_n \cap H_m$ are infinite subsets of $\omega$ by the assumptions of the lemma. So pick $k_1 \in Z_1 \cap \ldots Z_n \cap H_1$ such that $k_1 > k$ and $k_1 > l$. Also pick $k_2 \in Z_1 \cap \ldots Z_n \cap H_2$ such that $k_2 > k$ and $k_2 > l$ and (after finitely many steps, so we avoid choice here) $k_m \in Z_1 \cap \ldots Z_n \cap H_m$ with $k_m > k$ and $k_m >l$. Then $s:=\{k_1, k_2,\ldots, k_m\} \in J$ and it's clear that: $s \in \hat{Z_i}$ for all $i=1,\ldots,n$ and $s \in \tilde{H}_j$ for all $j \in \{1,\ldots,m\}$ and $s \in T_l$, all by construction. And because we ensured that the minimum of $s$ is beyond "my" $k$ as well, we see that $I$ is infinite (as a finite number of $s$ in $I$ could never all lie beyond any arbitrary bound, consider the max of their maxima which exists when all $s$ are finite, as is the case).

I think that's the (admittedly hidden) reason why we consider the finite conditions $s \in J$ (they're much easier to reason with) instead of infinite ones only, say.

Solution 2:

The reason why using $J = [\omega]^{<\omega} - \{\emptyset\}$ is necessary is because it is countable infinite. Recall that $\mathfrak{p}$ is defined such that if $\mathcal{F} \subseteq [\omega]^\omega$ with SFIP and $|\mathcal{F}|< \mathfrak{p}$ then $\mathcal{F}$ has a pseudo-intersection, i.e. it is defined for a family of countably infinite subsets of $\omega$. We define $J$ so that it is countably infinite and so we are actually allowed to use the definition of $\mathfrak{p}$. Because $|\omega| = |J|$ we may use a bijection to apply the definition of $\mathfrak{p}$ and find such a pseudo-intersection for $\mathcal{F}^*$. If we defined $J= \mathcal{P}(\omega)$ then $|J| = 2^{\aleph_0}$ and we cannot apply the definition of $\mathfrak{p}$.