Generalized tower numbers at uncountable cardinals: the exact consistency strength of $\mathfrak{t}_\kappa>\kappa^+$ and the $\pi$-depth equality $\pi\mathfrak{t}(U)=\pi\mathfrak{p}(U)$

For a regular uncountable cardinal $\kappa$, the generalized tower number $\mathfrak{t}_\kappa$ is the least length of a $\supseteq^*$-decreasing sequence of $\kappa$-sized subsets of $\kappa$ that has the $\kappa$-strong intersection property and no pseudo-intersection; it satisfies $\kappa^+\leq\mathfrak{p}_\kappa\leq\mathfrak{t}_\kappa\leq\mathfrak{b}_\kappa$. Benhamou and Goldberg (2025) proved that if $\kappa$ is measurable and $\mathfrak{t}_\kappa>\kappa^+$, then some inner model has a $\mu$-measurable cardinal, while Zapletal's equiconsistency of $\mathfrak{s}_\kappa>\kappa^+$ with Mitchell order $o(\kappa)=\kappa^{++}$ gives the lower bound $o(\kappa)=\kappa^{++}$ without measurability; the best upper bound is a supercompact cardinal, via a Mathias iteration that forces $\mathfrak{p}_\kappa=\mathfrak{t}_\kappa=\kappa^{++}$. The primary open problem, Question 4.1 of that paper, asks for the exact consistency strength of $\mathfrak{t}_\kappa>\kappa^+$ for regular $\kappa>\omega$: whether $o(\kappa)=\kappa^{++}$ suffices, and whether the $\mu$-measurable lower bound is optimal when $\kappa$ is required measurable. The same paper asks (Question 2.15) whether the $\pi$-depth analogue of the known equality $\mathfrak{t}(U)=\mathfrak{p}(U)$ holds for a uniform ultrafilter $U$ on $\kappa$, that is, whether $\pi\mathfrak{t}(U)=\pi\mathfrak{p}(U)$; the $\pi P$-point property entering this question is precisely the mechanism behind the $\mu$-measurable lower bound. Neither question is known to be resolved.

Background

Two of the classical cardinal characteristics of the continuum are the pseudo-intersection number $\mathfrak{p}$ and the tower number $\mathfrak{t}$: $\mathfrak{p}$ is the least size of a family of infinite subsets of $\omega$ whose finite subfamilies all have infinite intersections but which has no infinite set almost contained in every member, and $\mathfrak{t}$ is the least length of a $\supseteq^*$-decreasing sequence of infinite sets (a tower) with no such pseudo-intersection, where $A\supseteq^*B$ means that $B\setminus A$ is finite. A deep theorem of Malliaris and Shelah (published 2015) shows that $\mathfrak{p}=\mathfrak{t}$.

At a regular uncountable cardinal $\kappa$ the same definitions are read modulo $<\kappa$: for $A,B\subseteq\kappa$, $A\subseteq^*B$ means $|A\setminus B|<\kappa$, and $[\kappa]^\kappa$ denotes the family of $\kappa$-sized subsets of $\kappa$. A family $\mathcal{A}\subseteq[\kappa]^\kappa$ has the $\kappa$-strong intersection property ($\kappa$-SIP) if every subfamily of size $<\kappa$ has intersection of size $\kappa$; a pseudo-intersection of $\mathcal{A}$ is an $X\in[\kappa]^\kappa$ with $X\subseteq^*A$ for every $A\in\mathcal{A}$. A tower is a $\supseteq^*$-decreasing sequence $\langle A_i\mid i<\lambda\rangle\subseteq[\kappa]^\kappa$ with the $\kappa$-SIP and no pseudo-intersection; $\mathfrak{p}_\kappa$ is the least size of a family with the $\kappa$-SIP but no pseudo-intersection, $\mathfrak{t}_\kappa$ the least length of a tower, and Brooke-Taylor, Fischer, Friedman and Montoya (2017) proved $\kappa^+\leq\mathfrak{p}_\kappa\leq\mathfrak{t}_\kappa\leq\mathfrak{b}_\kappa$, where $\mathfrak{b}_\kappa$ is the unboundedness number of $\kappa^\kappa$ under eventual domination. (The SIP clause matters: without it, countable pseudo-towers exist by decomposing $\kappa$ into countably many $\kappa$-sized pieces, and different papers handle this either by requiring the SIP, as here, or by restricting tower lengths to regular $\lambda\geq\kappa$; the two conventions agree at all regular lengths $\geq\kappa$, because SIP is automatic for $\supseteq^*$-decreasing sequences of such lengths.) Whether $\mathfrak{p}_\kappa=\mathfrak{t}_\kappa$ is itself open; Fischer, Montoya, Schilhan and Soukup (2022) showed that either the equality holds or there is a strong gap.

The question concerns consistency strength: two set-theoretic statements are equiconsistent if one is consistent exactly when the other is, and such strength is calibrated by large cardinals. A cardinal $\kappa$ is measurable if it is the critical point of a nontrivial elementary embedding $j:V\to M$ of the universe into a transitive class; the Mitchell order $o(\kappa)$ measures how many such embeddings coexist on $\kappa$; $\kappa$ is $\mu$-measurable if for some such $j$ the normal measure derived from $j$ on $\kappa$, namely $\{X\subseteq\kappa\mid\kappa\in j(X)\}$, belongs to $M$; $\kappa$ is $\lambda$-strong if some $j:V\to M$ with critical point $\kappa$ has $V_\lambda\subseteq M$; and $\kappa$ is supercompact if such embeddings with closure exist at every $\lambda$. These hypotheses form a strict hierarchy in consistency strength, and when the core model $K$ — the canonical inner model without large cardinals — exists, large-cardinal strength in $V$ must be reflected in $K$.

Zapletal (1997), refined by Ben-Neria and Gitik (2015), showed that $\mathfrak{s}_\kappa>\kappa^+$ for regular $\kappa$ — where $\mathfrak{s}_\kappa$ is the splitting number, the least size of a family of subsets of $\kappa$ splitting every $\kappa$-sized set — is equiconsistent with $o(\kappa)=\kappa^{++}$. Since $\mathfrak{t}_\kappa\leq\mathfrak{s}_\kappa$, this yields a lower bound for $\mathfrak{t}_\kappa>\kappa^+$. Benhamou and Goldberg (2025) proved (their Theorem 3.14) that if $\kappa$ is measurable and $\mathfrak{t}_\kappa>\kappa^+$, then there is an inner model with a $\mu$-measurable cardinal, a jump above $o(\kappa)=\kappa^{++}$. On the upper-bound side, their paper notes that an indestructible supercompact $\kappa$ with a $<\kappa$-support iteration of Mathias forcing (a forcing adding a pseudo-intersection to suitable filters) of length $\kappa^{++}$ forces $\mathfrak{p}_\kappa=\mathfrak{t}_\kappa=\kappa^{++}$, and Honzik and Stejskalová (2025) realized $\kappa^+<\mathfrak{t}(\kappa)=\mathfrak{u}(\kappa)<2^\kappa$ at an inaccessible $\kappa$ from a supercompact with a weakly compact cardinal above. The exact strength of $\mathfrak{t}_\kappa>\kappa^+$ between these bounds is unknown.

The second question lives on the ultrafilter side. A uniform ultrafilter $U$ on $\kappa$ is an ultrafilter all of whose elements have size $\kappa$; its character $\mathfrak{ch}(U)$ is the least size of a base for $U$, and $\mathfrak{u}_\kappa$ is the minimum character over uniform ultrafilters. $U$ is a $P_\lambda$-point if every subfamily of $U$ of size $<\lambda$ has a $\supseteq^*$-lower bound inside $U$, and $\mathfrak{p}(U)=\min\{\lambda\mid U\text{ is not a }P_{\lambda^+}\text{-point}\}$; the depth spectrum $\Sp_{\dpt}(U)$, introduced in recent work of Benhamou, is the set of regular $\lambda$ admitting a $\subseteq^*$-decreasing sequence from $U$ with no $\subseteq^*$-lower bound in $U$, and $\mathfrak{t}(U)=\min\Sp_{\dpt}(U)$. For ultrafilters it is known that $\mathfrak{t}(U)=\mathfrak{p}(U)$. The $\pi$-analogues relax the lower-bound requirement from "in $U$" to "in $[\kappa]^\kappa$": $\pi\mathfrak{t}(U)=\min\Sp_{\pi\dpt}(U)$, where $\Sp_{\pi\dpt}(U)$ collects the regular $\lambda$ admitting a $\subseteq^*$-decreasing sequence from $U$ with no lower bound in $[\kappa]^\kappa$ at all, and $\pi\mathfrak{p}(U)=\min\{\lambda\mid U\text{ is not a }\pi P_{\lambda^+}\text{-point}\}$, where $U$ is a $\pi P_\lambda$-point if every subfamily of $U$ of size $<\lambda$ has a pseudo-intersection in $[\kappa]^\kappa$. Benhamou and Goldberg prove $\mathfrak{t}(U)\leq\pi\mathfrak{p}(U)\leq\pi\mathfrak{t}(U)\leq\pi\mathfrak{ch}(U)\leq\mathfrak{ch}(U)$ and ask whether the first $\pi$-inequality can be strict. The two questions are technically linked: the proof of the $\mu$-measurable lower bound shows that a normal ultrafilter on a measurable $\kappa$ must be a $\pi P_{\kappa^{++}}$-point, since a counterexample sequence could otherwise be refined to a tower of length $\kappa^+$, contradicting $\mathfrak{t}_\kappa>\kappa^+$.

Problem Statement

Both questions below are asked by Benhamou and Goldberg in "Measures that violate the generalized continuum hypothesis" (their Questions 4.1 and 2.15). The common objective is to complete the basic ZFC and consistency-strength picture of generalized tower numbers — globally at a regular uncountable cardinal, and in their $\pi$-depth form at a uniform ultrafilter.

Primary (Question 4.1). Determine the exact consistency strength of $\mathfrak{t}_\kappa>\kappa^+$ for a regular cardinal $\kappa>\omega$, where $\mathfrak{t}_\kappa$ is the least length of a $\supseteq^*$-decreasing tower in $[\kappa]^\kappa$ with the $\kappa$-strong intersection property and no pseudo-intersection (equivalently: there is no tower of length $\leq\kappa^+$). Concretely, close the gap between the known bounds:

(a) Plain case: is $\mathfrak{t}_\kappa>\kappa^+$ equiconsistent with $o(\kappa)=\kappa^{++}$ — the lower bound obtained from Zapletal's equiconsistency of $\mathfrak{s}_\kappa>\kappa^+$ with $o(\kappa)=\kappa^{++}$ via $\mathfrak{t}_\kappa\leq\mathfrak{s}_\kappa$ — or does its strength lie strictly between $o(\kappa)=\kappa^{++}$ and a supercompact cardinal, the current upper bound coming from an indestructible supercompact with a $<\kappa$-support Mathias iteration of length $\kappa^{++}$ that forces $\mathfrak{p}_\kappa=\mathfrak{t}_\kappa=\kappa^{++}$?

(b) Measurable case: Benhamou-Goldberg's Theorem 3.14 derives an inner model with a $\mu$-measurable cardinal from a measurable $\kappa$ with $\mathfrak{t}_\kappa>\kappa^+$. Is this optimal? That is, identify the weakest hypothesis $H$ such that $\mathrm{Con}(H)$ is equivalent to the consistency of "$\kappa$ is measurable and $\mathfrak{t}_\kappa>\kappa^+$"; in particular, determine whether a $\mu$-measurable cardinal, or any hypothesis strictly below supercompactness, suffices for the consistency of the statement.

Secondary (Question 2.15). For a uniform ultrafilter $U$ on a regular uncountable $\kappa$, is $\pi\mathfrak{t}(U)=\pi\mathfrak{p}(U)$? Here $\pi\mathfrak{t}(U)=\min\Sp_{\pi\dpt}(U)$, where $\Sp_{\pi\dpt}(U)$ is the set of regular $\lambda$ for which there is a $\subseteq^*$-decreasing sequence $\langle X_i\mid i<\lambda\rangle\subseteq U$ with no $\subseteq^*$-lower bound in $[\kappa]^\kappa$; and $\pi\mathfrak{p}(U)=\min\{\lambda\mid U\text{ is not a }\pi P_{\lambda^+}\text{-point}\}$, where $U$ is a $\pi P_\lambda$-point if every subfamily of $U$ of size $<\lambda$ has a pseudo-intersection in $[\kappa]^\kappa$. The known chain is $\mathfrak{t}(U)\leq\pi\mathfrak{p}(U)\leq\pi\mathfrak{t}(U)\leq\pi\mathfrak{ch}(U)\leq\mathfrak{ch}(U)$ and $\mathfrak{t}(U)=\mathfrak{p}(U)$ is known, so the question is whether the inequality $\pi\mathfrak{p}(U)\leq\pi\mathfrak{t}(U)$ is ever strict.

The two components are linked: the $\pi P$-point property of ultrafilters is exactly the mechanism by which the $\mu$-measurable lower bound of Theorem 3.14 is proved, so progress on the secondary question feeds directly into the primary one.

References

1. Tom Benhamou and Gabriel Goldberg, "Measures that violate the generalized continuum hypothesis", arXiv preprint arXiv:2503.20094 (2025), https://arxiv.org/abs/2503.20094 (v2 of 9 December 2025)
2. Tom Benhamou and Gabriel Goldberg, "Measures that violate the generalized continuum hypothesis", Advances in Mathematics 487:110778 (2026), DOI 10.1016/j.aim.2025.110778, https://doi.org/10.1016/j.aim.2025.110778
3. Jindřich Zapletal, "Splitting number at uncountable cardinals", The Journal of Symbolic Logic 62(1):35-42 (1997), DOI 10.2307/2275730, https://doi.org/10.2307/2275730
4. Omer Ben-Neria and Moti Gitik, "On the splitting number at regular cardinals", The Journal of Symbolic Logic 80(4):1348-1360 (2015), DOI 10.1017/jsl.2015.22, https://doi.org/10.1017/jsl.2015.22
5. A. D. Brooke-Taylor, V. Fischer, S. D. Friedman and D. C. Montoya, "Cardinal characteristics at κ in a small u(κ) model", Annals of Pure and Applied Logic 168(1):37-49 (2017), DOI 10.1016/j.apal.2016.08.004, https://doi.org/10.1016/j.apal.2016.08.004
6. Vera Fischer, Diana Carolina Montoya, Jonathan Schilhan and Dániel T. Soukup, "Towers and gaps at uncountable cardinals", Fundamenta Mathematicae 257(2):141-166 (2022), DOI 10.4064/fm109-9-2021, https://doi.org/10.4064/fm109-9-2021
7. Vera Fischer and Silvan Horvath, "Generalized tower spectra", The Journal of Symbolic Logic (2025), 1-17, DOI 10.1017/jsl.2024.61, https://doi.org/10.1017/jsl.2024.61
8. Radek Honzik and Šárka Stejskalová, "Generalized cardinal invariants for an inaccessible κ with compactness at κ++", Archive for Mathematical Logic 64(7-8):1077-1102 (2025), DOI 10.1007/s00153-025-00977-2, https://doi.org/10.1007/s00153-025-00977-2
9. Maryanthe Malliaris and Saharon Shelah, "Cofinality spectrum theorems in model theory, set theory, and general topology", Journal of the American Mathematical Society 29(1):237-297 (published online 2015), DOI 10.1090/jams830, https://doi.org/10.1090/jams830
10. Tom Benhamou, James Cummings, Gabriel Goldberg, Yair Hayut and Alejandro Poveda, "Isomorphism Classes of Generating Sets", arXiv preprint arXiv:2504.18381 (2025), https://arxiv.org/abs/2504.18381
11. Tom Benhamou and Corey Bacal Switzer, "Eventual Capture on a Measurable Cardinal", arXiv preprint arXiv:2511.07411 (2025), https://arxiv.org/abs/2511.07411
12. Tom Benhamou, Natasha Dobrinen and Tan Özalp, "Tukey-idempotency and strong p-points", arXiv preprint arXiv:2511.01050 (2025), https://arxiv.org/abs/2511.01050
13. Tom Benhamou, "On Ultrapowers and Cohesive Ultrafilters", arXiv preprint arXiv:2410.06275 (2024), https://arxiv.org/abs/2410.06275