# 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 $$\hat{\mathfrak t}$$?

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 $\hat t$? – Joel David Hamkins Dec 28 '19 at 10:07
• @JoelDavidHamkins Yes, I had in mind sup, not min (min yields the well-known cardinal $t=p$). – Taras Banakh Jan 1 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? – Taras Banakh Jan 2 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. – Joel David Hamkins Jan 2 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. – Taras Banakh Jan 5 at 9:05