The pseudointersection and tower numbers

In ZFC, with $[\omega {]}^{\omega }$, the strong finite intersection property, ${\subseteq }^{\ast }$ and towers as in Almost inclusion, pseudointersections and towers, define:

The pseudointersection number. $p$ is the least cardinality $\mid \mathcal{F}\mid$ of a family $\mathcal{F}\subseteq [\omega {]}^{\omega }$ that has the strong finite intersection property and has no pseudointersection. The collection of candidate cardinalities is nonempty: by A tower of size at most the continuum exists there is a tower, and a tower is a family with the strong finite intersection property, because a finite intersection $A_{{\beta }_0}\cap \cdots \cap A_{{\beta }_n}$ with ${\beta }_0<\cdots <{\beta }_n$ contains $A_{{\beta }_n}$ minus the union of the finitely many finite sets $A_{{\beta }_n}\setminus A_{{\beta }_i}$, and $A_{{\beta }_n}$ is infinite. The collection is a set of ordinals bounded by the cardinality of $[\omega {]}^{\omega }$, and each of its members is a cardinal (A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used), so the Axiom of Choice, which well-orders every subset of $[\omega {]}^{\omega }$ (The Axiom of Choice, The well-ordering theorem) and makes cardinality available, gives $p$ as the least element of that set; the minimum is attained, so there is an SFIP family of size $p$ with no pseudointersection.

The tower number. $t$ is the least ordinal $\lambda$ for which there is a tower of length $\lambda$, that is, a tower $⟨A_{\alpha }:\alpha <\lambda ⟩$. By A tower of size at most the continuum exists there is such a tower of some length ${\lambda }_0\le \mathfrak{c}$; take the least member of the set of qualifying ordinals $\lambda \le {\lambda }_0$. Thus $t$ exists, is attained and satisfies $t\le \mathfrak{c}$. No assertion that every possible tower length is at most $\mathfrak{c}$ is needed.

A shortest tower has cardinal length. The empty sequence is not a tower: $\omega$ is its pseudointersection. Nor can a tower have successor length $\beta +1$, because its last member $A_{\beta }$ is almost contained in every earlier member and is itself an infinite pseudointersection. Hence $t$ is a limit ordinal. Put $\rho =\mathrm{cf}(t)$ (Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals). There is a strictly increasing cofinal map $f:\rho \to t$ (For every ordinal $\alpha$ there is a least ordinal $\beta$ admitting a map $\beta \to \alpha$ with cofinal range, and that map may always be taken strictly increasing). Restrict a tower of length $t$ to the indices $f[\rho ]$: by the cofinal-subsequence argument in Almost inclusion, pseudointersections and towers, this remains a tower and has length exactly $\rho$. Minimality gives $t\le \rho$, while $\rho \le t$ by $\mathrm{cf}(\alpha )\le \alpha$; $\mathrm{cf}(0)=0$ and $\mathrm{cf}(\alpha +1)=1$; for a limit ordinal $\lambda$ the value $\mathrm{cf}(\lambda )$ is an infinite cardinal with $\mathrm{cf}(\mathrm{cf}(\lambda ))=\mathrm{cf}(\lambda )$, so it is regular; and every cofinal subset of $\lambda$ has cardinality at least $\mathrm{cf}(\lambda )$, a value that is attained, so $t=\mathrm{cf}(t)$. Since the cofinality of a limit ordinal is an infinite cardinal by that theorem, $t$ is in fact a regular infinite cardinal.

Normal form of a shortest tower. Removing repetitions as in Almost inclusion, pseudointersections and towers replaces a tower of length $\lambda$ by a strictly decreasing tower of order type $\theta \le \lambda$; only its cardinality is bounded by the number of almost-equality classes. Applied to $\lambda =t$, minimality gives $\theta =t$, so:

• there is a tower $⟨A_{\alpha }:\alpha <t⟩$ that is strictly decreasing, that is, $A_{\beta }{\supseteq }^{\ast }{A}_{\alpha }$ and $A_{\alpha }{\mathrm{\ne }}^{\ast }{A}_{\beta }$ whenever $\beta <\alpha <t$;
• $t\le 2^{{\mathrm{\aleph }}_0}=\mathfrak{c}$, by the existence construction above;
• $t$ is a regular cardinal, by the cofinal-subsequence argument above.

Equivalently, $t$ is the least number of distinct members in a tower. The strictly decreasing tower of length $t$ has exactly $t$ members. Conversely, if a tower has member set $S$, removal of repeats gives a tower of order type $\theta$ with $\mid \theta \mid \le \mid S\mid$. Minimality gives $t\le \theta$ as ordinals; because $t$ is an initial cardinal, this implies $t\le \mid \theta \mid \le \mid S\mid$ as cardinals. Thus no tower has fewer than $t$ distinct members.

Convention. When using a shortest tower below, take the strictly decreasing normal form; its length and the size of its member set both equal $t$. Monk states the pseudointersection number as $p=\min \{\mid \mathcal{F}\mid :\mathcal{F}\subseteq [\omega {]}^{\omega }$ has SFIP and no pseudo-intersection$\}$ and the tower number as the smallest ordinal that is the length of a tower; Malliaris and Shelah work with the forcing $P(\omega )\mathrm{/}\mathrm{f}\mathrm{i}\mathrm{n}$, where these same numbers are the standard cardinal characteristics of the almost-inclusion order.

The two definitions are not symmetric in the choice they consume. The minimum defining $p$ is a least cardinality of a set; AC makes cardinalities available for its candidate families. Once one tower has been constructed in ZFC, finding the least tower length among ordinals below that witness needs no additional choice. The cofinal-subsequence and repetition arguments establish that this least ordinal is the cardinal invariant used in the bounds below.

The inequality $p\le t$ is immediate from the two definitions and is proved with the remaining bounds in Basic bounds for p and t: a tower is an SFIP family with no pseudointersection, so the least size of such a family is at most the length of any tower, in particular at most $t$.