On Some Configurations Related to the Shelah Weak Hypothesis

The Keisler-Shelah isomorphism theorem and the continuum hypothesis

We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^ω/ \mathcal{U} \simeq \mathbf{N}^ω/ \mathcal{U}$ for some ultrafilter $\mathcal{U}$ on $ω,$ then $CH$ holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of $CH$.

The Keisler-Shelah isomorphism theorem and the continuum hypothesis II

We continue the investigation started in [Sh:1215] about the relation between the Keilser-Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequence $\mathbf m=\langle (\mathbb{M}^{1}_n, \mathbb{M}^{2}_n: n < ω\rangle$ of models of size at most $\aleph_1$ in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter $\mathcal D$ on $ω$, if the ultraproducts $\prod\limits_{\mathcal D} \mathbb{M}^{1}_n$ and $\prod\limits_{\mathcal D} \mathbb{M}…

ZFC中的一个Shelah群

A Shelah group in ZFC

本文仅基于ZFC公理系统（不依赖连续统假设）成功构造了Shelah群，提出了源自负方括号划分关系等理论的组合条件，并证明了其对正则基数后继成立，同时给出了极限基数大小的首个一致示例。

A note on "Another ordering of the ten cardinal characteristics in Cichoń's Diagram" and further remarks

In this note, we relax the hypothesis of the main results in Kellner-Shelah-Tǎnasie's "Another ordering of the ten cardinal characteristics in Cichoń's diagram".

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for pairs of cardinals $(\n,ρ)$ in which $n$ is finite and $ρ$ is infinite. An extension of Miller's theorem is proved here in ZFC for pairs of cardinals $(ν,ρ)$ in which $ν$ is arbitrary and $ρ\ge \beth_\om(ν)$. The proof uses a new general method that is based on Shelah's revises Generalized Continuum Hypothesis theorem. Upper bounds on conflict-free coloring numbers of families of sets and a general comparison theorem follow as corollaries of the main theorem. Other corollaries eliminate the use of additional axioms from splitting theorems due to E…

Distributive Aronszajn trees

Ben-David and Shelah proved that if $λ$ is a singular strong-limit cardinal and $2^λ=λ^+$, then $\square^*_λ$ entails the existence of a normal $λ$-distributive $λ^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_λ$ by $\square(λ^+,{<}λ)$. As $\square(λ^+,{<}λ)$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocess…