Golshani proved, from an increasing ω\omega-sequence of supercompact cardinals satisfying an additional normal-measure condition, that there is a model of ZFC containing a set BB of regular cardinals whose iterated pcf-sequence ⟨pcf⁡α(B):α∈Ord⟩\langle\operatorname{pcf}^{\alpha}(B):\alpha\in\mathrm{Ord}\rangle is strictly increasing — a consistent negative answer to a question of Tsukuura on whether ZFC forces every pcf-iteration to eventually stabilize. The exact consistency strength of this statement is unknown: the only lower bound known is that it contradicts GCH, and no construction from weaker large cardinals exists. The problem is to calibrate this statement in the hierarchy of hypotheses beyond ZFC, via constructions from weaker hypotheses and/or provable lower bounds on the necessity of large cardinals.

Background

Pcf (possible cofinalities) theory, developed by Shelah in his book Cardinal arithmetic, is the main tool for proving bounds in ZFC on powers of singular cardinals — for example, if ℵω\aleph_\omega is a strong limit cardinal (meaning 2λ<ℵω2^\lambda<\aleph_\omega for every λ<ℵω\lambda<\aleph_\omega) then 2ℵω<ℵω42^{\aleph_\omega}<\aleph_{\omega_4}. For a set AA of regular cardinals, pcf⁡(A)\operatorname{pcf}(A) is the set of cofinalities of the reduced products ∏A/D\prod A/D of AA modulo ultrafilters DD on AA; write Reg\mathrm{Reg} for the class of regular cardinals and, following the notation of these papers, η∩Reg\eta\cap\mathrm{Reg} for the set of regular cardinals below η\eta. When AA is a progressive interval of regular cardinals (an interval with min⁡(A)>∣A∣\min(A)>|A|), the operator is tame: pcf⁡(A)\operatorname{pcf}(A) is again an interval of regular cardinals, it has a largest element, its size is bounded in terms of ∣A∣|A|, and the operator is idempotent, pcf⁡(pcf⁡(A))=pcf⁡(A)\operatorname{pcf}(\operatorname{pcf}(A))=\operatorname{pcf}(A) (Shelah, Cardinal arithmetic). Outside the progressive setting this fails badly. Tsukuura proved that if κ\kappa is a measurable cardinal (an uncountable cardinal carrying a κ\kappa-complete nonprincipal ultrafilter), then pcf⁡(κ∩Reg)\operatorname{pcf}(\kappa\cap\mathrm{Reg}) is the set of all regular cardinals below (2κ)+(2^\kappa)^+, so a single application of the operator blows the regulars below κ\kappa up into a huge non-progressive set. He also observed that under the Generalized Continuum Hypothesis (GCH) the operator is idempotent on every set, pcf⁡(pcf⁡(A))=pcf⁡(A)\operatorname{pcf}(\operatorname{pcf}(A))=\operatorname{pcf}(A) for all A⊆RegA\subseteq\mathrm{Reg}, and that under GCH with no weakly inaccessible cardinals pcf⁡\operatorname{pcf} admits a completely explicit description.

One can iterate the operator through the ordinals: pcf⁡0(A)=A\operatorname{pcf}^{0}(A)=A, pcf⁡α+1(A)=pcf⁡(pcf⁡α(A))\operatorname{pcf}^{\alpha+1}(A)=\operatorname{pcf}(\operatorname{pcf}^{\alpha}(A)), and pcf⁡λ(A)=⋃β<λpcf⁡β(A)\operatorname{pcf}^{\lambda}(A)=\bigcup_{\beta<\lambda}\operatorname{pcf}^{\beta}(A) for limit λ\lambda. Since A⊆pcf⁡(A)A\subseteq\operatorname{pcf}(A), this sequence is increasing, and Tsukuura asked whether ZFC proves that it always stabilizes, i.e. whether for every A⊆RegA\subseteq\mathrm{Reg} some α\alpha has pcf⁡α+1(A)=pcf⁡α(A)\operatorname{pcf}^{\alpha+1}(A)=\operatorname{pcf}^{\alpha}(A). From an increasing ω\omega-sequence of supercompact cardinals (a supercompact cardinal κ\kappa is one admitting elementary embeddings j:V→Mj:V\to M with critical point κ\kappa and MM closed under λ\lambda-sequences for arbitrarily large λ\lambda) he had already produced a forcing extension in which, for A=κ0∩RegA=\kappa_0\cap\mathrm{Reg}, the finite iterates ⟨pcf⁡n(A):n<ω⟩\langle\operatorname{pcf}^{n}(A):n<\omega\rangle are strictly increasing.

Golshani gave a consistent negative answer to Tsukuura's question. Assuming an increasing sequence ⟨κn:n<ω⟩\langle\kappa_n:n<\omega\rangle of supercompact cardinals whose first element κ0\kappa_0 additionally satisfies that the set A={α<κ0:α is <κ0-supercompact}A=\{\alpha<\kappa_0:\alpha\text{ is }{<}\kappa_0\text{-supercompact}\} has measure one for every normal measure on κ0\kappa_0 (a normal measure is a κ\kappa-complete ultrafilter closed under diagonal intersections), he constructs a model MM containing an inaccessible cardinal θ\theta such that in MM the sequence ⟨pcf⁡α(θ∩Reg):α∈Ord⟩\langle\operatorname{pcf}^{\alpha}(\theta\cap\mathrm{Reg}):\alpha\in\mathrm{Ord}\rangle is strictly increasing. The construction is a three-step forcing: Laver's indestructibility preparation (a preliminary forcing making supercompactness survive further directed-closed forcing), augmented by the Apter–Hamkins universal indestructibility preparation so that the members of AA keep their partial supercompactness; full-support products of Cohen forcings Add⁡(κn,κn+1)\operatorname{Add}(\kappa_n,\kappa_{n+1}) prescribing the continuum function, 2κn=κn+12^{\kappa_n}=\kappa_{n+1}; and the generalized Radin forcing of Foreman and Woodin (the forcing from their proof that GCH can fail everywhere, which adds a closed unbounded set carrying large-cardinal structure so the process can continue at limit stages). The iterates are computed explicitly: at limit stages α\alpha (including α=0\alpha=0) one obtains the regular cardinals below the α\alpha-th point θα\theta_\alpha of the club, and at stages α=δ+n\alpha=\delta+n (δ\delta limit, n>0n>0) the regular cardinals below in(θδ)+i_n(\theta_\delta)^+, where the ini_n are the embedding-derived cardinals of the Foreman–Woodin generalized Radin forcing (for successive points α<β\alpha<\beta of the club, they satisfy i5(α)=2i4(α)=βi_5(\alpha)=2^{i_4(\alpha)}=\beta). He also notes that θ0\theta_0 may be taken (θ0+2)(\theta_0+2)-strong (an elementary-embedding property stronger than measurability) and that Magidor forcing — a Prikry-type forcing that changes the cofinality of a measurable cardinal — can make θ0\theta_0 singular of any prescribed regular cofinality η<θ0\eta<\theta_0 without destroying the conclusion.

This supplies the only upper bound on the consistency strength of the phenomenon. No lower bound beyond ZFC-level observations is known — the statement contradicts GCH, but ¬\negGCH itself carries no consistency strength — and no construction from weaker large cardinals is known.

Problem Statement

Determine the consistency strength of the statement

Φ  :  there is a set B⊆Reg such that ⟨pcf⁡α(B):α∈Ord⟩ is strictly increasing,\Phi\;:\;\text{there is a set }B\subseteq\mathrm{Reg}\text{ such that }\langle\operatorname{pcf}^{\alpha}(B):\alpha\in\mathrm{Ord}\rangle\text{ is strictly increasing},

that is, pcf⁡α(B)⊊pcf⁡α+1(B)\operatorname{pcf}^{\alpha}(B)\subsetneq\operatorname{pcf}^{\alpha+1}(B) for every ordinal α\alpha; here Reg\mathrm{Reg} is the class of regular cardinals, pcf⁡(A)\operatorname{pcf}(A) is the set of cofinalities of the reduced products ∏A/D\prod A/D modulo ultrafilters DD on AA, and the iterates are defined by pcf⁡0(B)=B\operatorname{pcf}^{0}(B)=B, pcf⁡α+1(B)=pcf⁡(pcf⁡α(B))\operatorname{pcf}^{\alpha+1}(B)=\operatorname{pcf}(\operatorname{pcf}^{\alpha}(B)), and pcf⁡λ(B)=⋃β<λpcf⁡β(B)\operatorname{pcf}^{\lambda}(B)=\bigcup_{\beta<\lambda}\operatorname{pcf}^{\beta}(B) for limit λ\lambda. At present Con⁡(ZFC+Φ)\operatorname{Con}(\mathrm{ZFC}+\Phi) is known to follow from an increasing ω\omega-sequence of supercompact cardinals whose first element κ0\kappa_0 satisfies that {α<κ0:α is <κ0-supercompact}\{\alpha<\kappa_0:\alpha\text{ is }{<}\kappa_0\text{-supercompact}\} has measure one for every normal measure on κ0\kappa_0 (a κ0\kappa_0-complete ultrafilter on κ0\kappa_0 closed under diagonal intersections), and Φ\Phi is known to fail under the Generalized Continuum Hypothesis (GCH); nothing else is known. The objective is to calibrate Φ\Phi in the hierarchy of hypotheses beyond ZFC.

References

  1. Golshani, Mohammad. 'Unlimited accumulation by Shelah's PCF operator'. Periodica Mathematica Hungarica 88 (2024), no. 1, pp. 273-280. DOI: 10.1007/s10998-023-00553-2 (arXiv:2205.10709). https://doi.org/10.1007/s10998-023-00553-2
  2. Tsukuura, Kenta. 'Prikry-type forcing and the set of possible cofinalities'. Tsukuba Journal of Mathematics 45 (2021), no. 2, pp. 83-95. DOI: 10.21099/tkbjm/20214502083 (arXiv:2201.02322). https://doi.org/10.21099/tkbjm/20214502083
  3. Shelah, Saharon. 'Cardinal arithmetic'. Oxford Logic Guides 29, Oxford: Clarendon Press, 1994. ISBN: 0-19-853785-9 (zbMATH: 0848.03025). https://zbmath.org/0848.03025
  4. Foreman, Matthew; Woodin, W. Hugh. 'The generalized continuum hypothesis can fail everywhere'. Annals of Mathematics (2) 133 (1991), no. 1, pp. 1-35. DOI: 10.2307/2944324. https://doi.org/10.2307/2944324
  5. Laver, Richard. 'Making the supercompactness of κ indestructible under κ-directed closed forcing'. Israel Journal of Mathematics 29 (1978), no. 4, pp. 385-388. DOI: 10.1007/BF02761175. https://doi.org/10.1007/BF02761175
  6. Magidor, Menachem. 'Changing cofinality of cardinals'. Fundamenta Mathematicae 99 (1978), no. 1, pp. 61-71. DOI: 10.4064/fm-99-1-61-71. https://doi.org/10.4064/fm-99-1-61-71
  7. Apter, Arthur W.; Hamkins, Joel David. 'Universal indestructibility'. Kobe Journal of Mathematics 16 (1999), no. 2, pp. 119-130. zbMATH: 0953.03060. https://zbmath.org/0953.03060