# Relations between two tower numbers

A tower is a subset $$T\subset [\omega]^\omega$$ of the family $$[\omega]^\omega$$ of all infinite subsets of $$\omega$$ such that $$T$$ is well-ordered by the relation $$\supset^*$$ of almost inclusion and has no infinite pseudointersections. A tower is regular if its cardinality is a regular cardinal.

Consider two small uncountable cardinals:

$$\mathfrak t=\min\{|T|:T\subset[\omega]^\omega$$ is a tower$$\}$$;

$$\hat{\mathfrak t}=\sup\{|T|:T\subset[\omega]^\omega$$ is a regular tower$$\}$$.

It is clear that $$\mathfrak t\le\hat{\mathfrak t}\le\mathfrak c.$$ Martin's Axiom implies $$\mathfrak t=\hat{\mathfrak t}=\mathfrak c$$. On the other hand the strict inequality $$\mathfrak t<\hat{\mathfrak t}$$ is known to be consistent (but I do not know the reference).

The cardinal $$\mathfrak t$$, called the tower number, is well-studied in Set Theory.

I am interested in the cardinal $$\hat{\mathfrak t}$$, more precisely in its relation to other known cardinal characteristics of the continuum.

Problem 1. Is there any non-trivial upper or lower bound for the cardinal $$\hat{\mathfrak t}$$? In particular, is $$\mathfrak b\le\hat{\mathfrak t}$$?

Problem 2. In which known models of ZFC does the strict inequality $$\mathfrak t<\hat{\mathfrak t}$$ hold?

Maybe there is some fixed standard notation for $$\hat{\mathfrak t}$$?

• Did you really mean $\sup$ and not $\min$ in $\hat t$? Dec 28 '19 at 10:07
• @JoelDavidHamkins Yes, I had in mind sup, not min (min yields the well-known cardinal $t=p$). Jan 1 '20 at 11:18

This is not a full answer, but I claim that the restriction to regular towers is not relevant.

Theorem. The cardinal $$\hat t$$ is also the supremum of all tower sizes, not just the regular towers. $$\hat t=\sup\{\ |T|\ \mid\ T\subset[\omega]^\omega\text{ is a regular tower }\}=\sup\{\ |T|\ \mid\ T\subset[\omega]^\omega\text{ is a tower }\}.$$

This will be a consequence of the following lemma.

Lemma. If $$T$$ is any tower and $$S\subset[\omega]^\omega$$ is any chain of sets well-ordered by $$\supset^*$$, then $$T$$ is a subtower of a tower $$T^*$$ with $$|T^*|\geq|S|$$.

Proof of lemma. We may assume $$T$$ is continuous, and so there are successive sets $$A$$ and $$B$$ appearing in $$T$$ with infinite difference $$A-B$$. We may build a copy of $$S$$ on this difference set, containing $$B$$, and place it between $$A$$ and $$B$$ in $$T$$, thereby constructing a new tower $$T^*$$ of size at least $$|S|$$. This new chain $$T^*$$ is a tower since it is still well-ordered by $$\supset^*$$ and since it agrees with $$T$$ on a final segment, it still has no infinite pseudointersection. $$\Box$$

Proof of theorem. Suppose that $$S$$ is any tower, and that $$T$$ is a regular tower. If $$|S|$$ is regular, then by a direct application of the lemma, we find a regular tower $$T^*$$ of size at least $$|S|$$. If $$|S|$$ is singular, then this is a limit of regular cardinals, and we may extend $$T$$ to various $$T'$$ of at least those regular sizes, by using only part of $$S$$ in the extending process. So the supremum of the sizes of the regular towers is the same as the supremum of the sizes of any tower. $$\quad\Box$$

I am not sure, however, how large the towers can be. (Without the well-ordered requirement, we can of course find towers of size continuum, using Dedekind cuts in the rationals.)

• Unfortunately, your proof is not clear to me. If you create a tower on the base of a tower $T$ inserting new sets, then you do not change the cofinality of that tower. Consequently, the cofinality of the new tower $T^*$ is equal to the cofinality of the tower $T$ and this does not produce a regular tower. Or I do not see something important? Jan 2 '20 at 9:05
• I agree, I have changed only the cardinality and not the cofinality of the tower. But you defined regular tower in terms of its cardinality, not cofinality. I found that strange, and this is why I posted this answer. Of course, the cofinality of a tower will always be a regular cardinal. Jan 2 '20 at 12:43
• Thank you for your comment due to which I have understood that we (with Bardyla) introduced a wrong defintion of $\hat t$. In that definition the cardinality should be changed by cofinality. Jan 5 '20 at 9:05

Assuming I understand the definitions correctly, I can give you a couple of references.

(1) Dordal (see below) gives a model in which $$\mathfrak{b}=\mathfrak{c}=\aleph_2$$ and all towers have cardinality $$\aleph_1$$. Thus, in his model $$\mathfrak{t}=\hat{\mathfrak{t}}=\aleph_1<\aleph_2=\mathfrak{b}=\mathfrak{c}$$, and so the last question in your first problem has a negative answer.

Edit: In fact, $$\mathfrak{h}=\aleph_2$$ in the model since he is using Mathias forcing, so I'm not sure of any good candidate for a non-trivial lower bound for $$\hat{\mathfrak{t}}.$$

(2) On the other hand, the model of Blass and Shelah discussed here gives a model in which $$\mathfrak{t}=\mathfrak{b}=\aleph_1$$ and there is a tower of length $$\aleph_2$$ (namely, the generating tower for the simple $$P_{\aleph_2}$$-point). Thus, this gives a model where $$\mathfrak{t}<\hat{\mathfrak{t}}$$.

Edit 3/17/20

I tracked down another reference (due to Dordal [2]) with some more information on this question. Theorem 1.3 in the paper gives a connection between towers in the structures $$^\omega\omega$$ and $$[\omega]^\omega$$:

Theorem (Dordal, Theorem 1.3 of [2])

Let $$\kappa$$ be an uncountable regular cardinal, and suppose there is a $$\kappa$$-tower in $$^\omega\omega$$ but not $$[\omega]^\omega$$. Then there is a $$\kappa$$-scale in $$^\omega\omega$$ (that is, $$\mathfrak{b}=\mathfrak{d}=\kappa$$).

With regard to Problem 1, we may say that if $$\mathfrak{b}<\mathfrak{d}$$ then $$\mathfrak{b}\leq\hat{\mathfrak{t}}$$ (apply the above theorem, and use the fact that there is a tower of length $$\mathfrak{b}$$ in the structure $$^\omega\omega$$. On the other hand, it is consistent that $$\mathfrak{b}=\mathfrak{d}=\aleph_2$$ and $$\hat{\mathfrak{t}}=\aleph_1$$ (this occurs in the model from [1]).

The paper has many other forcing constructions showing that not much more can be said.

[1] Dordal, Peter Lars, A model in which the base-matrix tree cannot have cofinal branches, J. Symb. Log. 52, 651-664 (1987). ZBL0637.03049.

[2] Dordal, Peter Lars, Towers in $$[\omega]^{\omega}$$ and $$^{\omega}\omega$$, Ann. Pure Appl. Logic 45, No. 3, 247-276 (1989). ZBL0686.03024.