Journal of Mathematical Logic

Extender-based Prikry–Magidor forcing for overlapping extenders is introduced. As an application, models with strong forms of negations of the Shelah Weak Hypothesis for various cofinalities are constructed.

Let ${𝒦}$ be an expansion of either an ordered field $(K,\le )$, or a valued field $(K,v)$. Given a definable set $X\subseteq K^m$ let ${𝒞}(X)$ be the ring of continuous definable functions from $X$ to $K$. Under very mild assumptions on the geometry of $X$ and on the structure ${𝒦}$, in particular when ${𝒦}$ is $o$-minimal or $P$-minimal, or an expansion of a local field, we prove that the ring of integers $\mathbb{Z}$ is interpretable in ${𝒞}(X)$. If ${𝒦}$ is $o$-minimal and $X$ is definably connected of pure dimension $\ge 2$, then ${𝒞}(X)$ defines the subring $\mathbb{Z}$. If ${𝒦}$ is $P$-minimal and $X$ has no isolated points, then there is a discrete ring ${𝒵}$ contained in $K$ and naturally isomorphic to $\mathbb{Z}$, such that the ring of functions $f\in {𝒞}(X)$ which take values in ${𝒵}$ is definable in ${𝒞}(X)$.

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain ($CAC$) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma ($WKL$), including showing that $CAC+WKL$ does not imply the thin set theorem for pairs, and that the principle “the product of well-quasi-orders is a well-quasi-order” is strictly between $CAC$ and the Ascending/Descending Sequences principle, even in the presence of $WKL$.

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_{\sigma }$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $\varnothing$. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932] which says that for any strong type defined on a single complete type over $\varnothing$, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups.

We show that it is consistent relative to a huge cardinal that for all infinite cardinals $\kappa$, ${\square }_{\kappa }$ holds and there is a stationary $S\subseteq {\kappa }^{+}$ such that ${\mathrm{\text{NS}}}_{{\kappa }^{+}}\upharpoonright S$ is ${\kappa }^{++}$-saturated.