Tootfinder

On Shelah's Approachability Ideal
Hannes Jakob, Alejandro Poveda
https://arxiv.org/abs/2508.04374 https://arxiv.org/pdf/2508.04374

On Shelah's Approachability Ideal
We solve a long-standing open problem of Shelah regarding the \emph{Approachability Ideal} $I[κ^+]$. Given a singular cardinal $\aleph_γ$, a regular cardinal $μ\in (\mathrm{cf}(γ),\aleph_γ)$ and assuming appropriate large cardinal hypotheses, we construct a model of $\mathsf{ZFC}$ in which $\aleph_{γ+1} \cap \mathrm{cof}(μ) \notin I[\aleph_{γ+1}]$. This provides a definitive answer to a question of Shelah from the 80's. In addition, assuming large cardinals, we construct a model of $\ma…

A $\sigma$-morphic convex protoset
Aleksa D\v{z}uklevski
https://arxiv.org/abs/2507.20867 https://arxiv.org/pdf/2507.20867…

A $σ$-morphic convex protoset
We say that a tile is $σ$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $σ$-morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is $σ$-morphic. The result is interesting since all the constructions of $σ$-morphic sets of tiles that arise in the literature make use of bumps and nicks, which necessarily make the tiles non-convex. We construct our set by cl…

Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
Mirna D\v{z}amonja
https://arxiv.org/abs/2509.19519 …

Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
We prove a theorem on iterated forcing that can be used for preservation of $\aleph_2$ and $\aleph_1$ in iterations with supports of size $\aleph_1$ of forcings that have amalgamation properties similar to those present in the perfect set forcing. The work is modelled after Baumgartner's Axiom A and his proof that iterations with countable support of the same preserve $\aleph_1$. In honour of James E. Baumgartner, the property introduced here is called Property B$(κ)$. The known additional dif…

A guide to constructing free transitive actions on median spaces
P\'en\'elope Azuelos
https://arxiv.org/abs/2507.22230 https://arxiv.org/pdf/2507.2…

A guide to constructing free transitive actions on median spaces
We construct large families of groups admitting free transitive actions on median spaces. In particular, we construct groups which act freely and transitively on the complete universal real tree with continuum valence such that any subgroup of the additive reals is realised as the stabiliser of an axis. We prove a more precise version of this, which implies that there are $2^{2^{\aleph_0}}$ pairwise non-isomorphic groups which admit a free transitive action on this real tree. We also construct …

Analysis note: measurement of thrust in $e^{ }e^{-}$ collisions at $\sqrt{s}$ = 91 GeV with archived ALEPH data
Anthony Badea, Austin Baty, Hannah Bossi, Yu-Chen Chen, Yi Chen, Jingyu Zhang, Gian Michele Innocenti, Marcello Maggi, Chris McGinn, Michael Peters, Tzu-An Sheng, Vinicius Mikuni, Matthew Avaylon, Patrick Komiske, Eric Metodiev, Jesse Thaler, Benjamin Nachman, Yen-Jie Lee

Analysis note: measurement of thrust in $e^{+}e^{-}$ collisions at $\sqrt{s}$ = 91 GeV with archived ALEPH data
A measurement of the thrust distribution in $e^{+}e^{-}$ collisions at $\sqrt{s} = 91.2$ GeV with archived data from the ALEPH experiment at the Large Electron-Positron Collider is presented. The thrust distribution is reconstructed from charged and neutral particles resulting from hadronic $Z$-boson decays. For the first time with $e^{+}e^{-}$ data, detector effects are corrected using a machine learning based method for unbinned unfolding. The measurement provides new input for resolving curr…

A Solovay-like model at $\aleph_\omega$
Alejandro Poveda, Sebastiano Thei
https://arxiv.org/abs/2509.18991 https://arxiv.org/pdf/2509.18991

A Solovay-like model at $\aleph_ω$
Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_ω$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_ω))$ satisfies the following properties: (1) Every set $A\subseteq (\aleph_ω)^ω$ has the $\aleph_ω$-PSP. (2) There is no scale at $\aleph_ω$. (3) The Singular Cardinal Hypothesis (SCH) fails at $\aleph_ω$. (4) Shelah's Approachability property (AP) fails at $\aleph_ω$. (5) The Tree Property (TP) holds at $\…

Ultrafilters over Successor Cardinals and the Tukey Order
Tom Benhamou, Justin T. Moore, Luke Serafin
https://arxiv.org/abs/2507.22307 https://arxiv.org/pd…

Ultrafilters over Successor Cardinals and the Tukey Order
We study ultrafilters on regular uncountable cardinals, with a primary focus on $ω_1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform ultrafilter over $ω_1$ is Tukey-equivalent to $[2^{\aleph_1}]^{<ω}$, and for each cardinal $κ$ of uncountable cofinality, a new construction of a uniform ultrafilter over $κ$ which extends the club filter and is Tukey-equivalent to $[2^κ]^{<ω}$. We also analyze…

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Cofinal families of finite VC-dimension
Omer Ben-Neria, Itay Kaplan, George Peterzil
https://arxiv.org/abs/2509.24744 https://arxiv.org/pdf/2509.24744

Cofinal families of finite VC-dimension
Given infinite cardinals $θ\leq κ$, we ask for the minimal VC-dimension of a cofinal family $\mathcal{F}\subseteq[κ]^{<θ}$. We show that for $θ=ω$ and $κ=\aleph_n$ it is consistent with ZFC that there exists such a family of VC-dimension $n+1$, which is known to be the lower bound. For $θ>ω$ we answer this question completely, demonstrating a strong dichotomy between the case of singular and regular $θ$. We furthermore answer some relative and generalized versions of the above questio…

Uniformization of ladder system colorings and stationary precaliber forcings
Yushiro Aoki
https://arxiv.org/abs/2508.18900 https://arxiv.org/pdf/2508.18900…

Uniformization of ladder system colorings and stationary precaliber forcings
We investigate the relationship between variants of the uniformization property for ladder system colorings and fragments of Martin's Axiom. The well-known forcing properties of having precaliber $\aleph_1$ and being $σ$-centered correspond to uncountable refinement and countable decomposition into centered subsets, respectively, and the associated forcing axioms have been widely studied. In this paper, we focus on a forcing axiom for the property corresponding to stationary refinement, namely…

No covering with nowhere dense \textsf{P}-sets in the Cohen model
Alan Dow, Osvaldo Guzm\'an
https://arxiv.org/abs/2507.22476 https://arxiv.org/pdf/250…

No covering with nowhere dense \textsf{P}-sets in the Cohen model
We prove that if less than $\aleph_ω$-many Cohen reals are added to a model of \textsf{CH}, then $ω^{\ast}$ can not be covered by nowhere dense \textsf{P}-sets (equivalently, there is an ultrafilter on $ω$ that does not contain a tower).

The number of countable models of first-order theories
Anand Pillay, Predrag Tanovi\'c
https://arxiv.org/abs/2508.06854 https://arxiv.org/pdf/2508.0685…

The number of countable models of first-order theories
Throughout, $T$ denotes a complete first-order theory in a countable language $L$ that has infinite models and $I(\aleph_0,T)$ denotes the number of countable models of $T$, up to an isomorphism. To determine $I(\aleph_0,T)$, it suffices to consider only countable models of $T$ with domain $ω$; since there are at most continuum many $L$-structures with domain $ω$, $I(\aleph_0,T)\leqslant 2^{\aleph_0}$ holds. Theories with $I(\aleph_0,T)=1$ are the $\aleph_0$-categorical theories. These includ…

A small ultrafilter number at all $\aleph_3\leq\kappa<\aleph_\omega$
Julian Eshkol
https://arxiv.org/abs/2508.07456 https://arxiv.org/pdf/2508.07456

A small ultrafilter number at all $\aleph_3\leqκ<\aleph_ω$
This note shows how recent work of Eskew and Hayut can be combined with a method of Raghavan and Shelah to show the consistency of $\mathfrak u_κ<2^κ$ for $\aleph_3\leq κ<\aleph_ω$, from a huge cardinal.

Degree of Kripke-incompleteness of Tense Logics
Qian Chen
https://arxiv.org/abs/2507.04533 https://arxiv.org/pdf/2507.04533

Degree of Kripke-incompleteness of Tense Logics
The degree of Kripke-incompleteness of a logic $L$ in some lattice $\mathcal{L}$ of logics is the cardinality of logics in $\mathcal{L}$ which share the same class of Kripke-frames with $L$. A celebrated result on Kripke-incompleteness is Blok's dichotomy theorem for the degree of Kripke-incompleteness in $\mathsf{NExt}(\mathsf{K})$: every modal logic $L\in\mathsf{NExt}(\mathsf{K})$ is of the degree of Kripke-incompleteness $1$ or $2^{\aleph_0}$. In this work, we show that the dichotomy theorem…

Ultrafilters in the random real model
Alan Dow, Osvaldo Guzm\'an
https://arxiv.org/abs/2507.08346 https://arxiv.org/pdf/2507.0834…

Ultrafilters in the random real model
We prove that \textsf{P}-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph_ω% $-many random reals to a model of \textsf{CH. }These results improve and correct previous theorems that can be found in the literature.