Journal of Mathematical Logic

If $\mathrm{\Gamma }$ is a closed Noetherian graph on a $\sigma$-compact Polish space with no infinite cliques, it is consistent with the choiceless set theory ZF$+$DC that $\mathrm{\Gamma }$ is countably chromatic and there is no Vitali set.

Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on $\omega$-h-minimality and $\ell$-h-minimality, for $\ell$ a positive integer. Our main results include an analytic criterion for $\ell$-h-minimality, preservation of $\ell$-h-minimality under coarsening of the valuation and $\ell$-dimensional geometry.

In this paper, we give a characterization of the strong degrees of categoricity of computable structures greater or equal to $0^{″}$. They are precisely the treeable degrees — the least degrees of paths through computable trees — that compute $0^{″}$. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree $d$ with $0^{(\alpha )}\le d\le 0^{(\alpha +1)}$ for $\alpha$ a computable ordinal greater than 2 is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree $d$ with $0^{\prime }\le d\le 0^{″}$ is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree $d$ with $0^{\prime }<d<0^{″}$ that is not the degree of categoricity of a rigid structure.

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length $\omega$ on $\kappa$ is equivalent to weak compactness. Winning the game of length $2^{\kappa }$ is equivalent to $\kappa$ being measurable. We show that for games of intermediate length $\gamma$, II winning implies the existence of precipitous ideals with $\gamma$-closed, $\gamma$-dense trees.

The second part shows the first is not vacuous. For each $\gamma$ between $\omega$ and ${\kappa }^{+}$, it gives a model where II wins the games of length $\gamma$, but not ${\gamma }^{+}$. The technique also gives models where for all ${\omega }_1<\gamma \le \kappa$ there are $\kappa$-complete, normal, ${\kappa }^{+}$-distributive ideals having dense sets that are $\gamma$-closed, but not ${\gamma }^{+}$-closed.

We construct a model in which MA${}_{{\omega }_1}(S)[S]$ holds and ${\mathcal{𝒦}}_2$ fails. This shows that MA${}_{{\omega }_1}(S)[S]$ does not imply ${\mathcal{𝒦}}_2$ and answers an old question of Larson and Todorcevic in [Katetov’s problem, Trans. Amer. Math. Soc. 354(5) (2002) 1783–1791]. We also investigate different strong colorings in models of MA${}_{{\omega }_1}(S)[S]$.

When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map $\text{res}:k((t))\to k$, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for t. Driven by this observation, we study the theory ${\text{VF}}_{\text{res},\iota }$ of valued fields equipped with a linear form $\text{res}:K\to k$ which restricts to the residue map on the valuation ring. We prove that ${\text{VF}}_{\text{res},\iota }$ does not admit a model companion. In addition, we show that $(k((t)),\text{res})$ is undecidable whenever k is an infinite field. As a consequence, we get that $(ℂ((t)),{\text{Res}}_0)$ is undecidable, where ${\text{Res}}_0:f\mapsto {\text{Res}}_0(f)$ maps f to its complex residue at 0.

We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation — can the filter of dense open subsets of a $\kappa$-distributive forcing notion of size $\kappa$ be extended to a $\kappa$-complete ultrafilter.