The generalized continuum hypothesis above a strongly compact cardinal

In analogy to the study of Scott rank/complexity of countable structures, we initiate the study of the Wadge degrees of the set of homeomorphic copies of topological spaces. One can view our results as saying that the classical characterizations of $[0,1]$ (e.g., as the unique continuum with exactly two non-cut points, and other similar characterizations), appropriated expressed, are the simplest possible characterizations of $[0,1]$. Formally, we show that the set of homeomorphic copies of $[0,1]$ is $\mathbfΠ^0_4$-Wadge-complete. We also show that the set of homeomorphic copies of $S^1$ is $\mathbfΠ^0_4$-Wadge-complete. On the other hand, we show that the set of homeomorphic copies of $\mathbb{R}$ is $\mathbfΠ^1_1$-Wadge-complete. It is the local compactness that cannot be expressed in a Borel way; the set of homeomorphic copies of $\mathbb{R}$ is $\mathbfΠ^0_4$-Wadge-complete within the locally compact spaces.

math.LO↗

arXiv2026

Loops, Inverse Limits and Non-Determinism

We introduce an operator on problems in Weihrauch complexity, which we call the infinite loop or inverse limit, and which corresponds to an infinite compositional product. This operation arises naturally whenever one implements algorithms that produce a sequence of results in an infinite loop, using some fixed subroutine. We prove that the corresponding operator is monotone with respect to (strong) Weihrauch reducibility but that it is not a closure operator. One of our findings is that weak KÅ‘nig's lemma is closed under infinite loops, which implies that the class of non-deterministically computable problems is also closed under this operation. Consequently, this class allows for a high degree of flexibility in programming. As our main technical tools, we present an injective version of the recursion theorem and an infinitary version of the so-called independent choice theorem. We also show that, in general, the infinite loop operator is more powerful than the composition of the diamond operator followed by the parallelization operator. However, in many practical scenarios, these compositions yield a result, which coincides with the application of the infinite loop operator. Finally, we discuss the special situation of loops for single-valued problems and for problems on Turing degrees.

math.LO↗

arXiv2026

The additivity of a certain Hausdorff measure can differ from that of the Lebesgue measure

Let $\mathcal{N}^h_Ω$ be the null ideal of the Hausdorff measure constructed by Davies and Rogers. We give a Tukey reduction of $(\mathcal{N}^h_Ω,\mathcal{N}^h_Ω,\subseteq)$ to a localization system with finite coordinate sets. It follows that $\mathfrak{v}^\forall_{D,g}\le\operatorname{add}(\mathcal{N}^h_Ω)$ and $\operatorname{cof}(\mathcal{N}^h_Ω)\le\mathfrak{c}^\forall_{D,g}$ for some $D, g \in ω^ω$. Consequently, we prove the consistency of $\mathfrak{d}<\operatorname{add}(\mathcal{N}^h_Ω)$ and, separately, $\operatorname{cof}(\mathcal{N}^h_Ω)<\mathfrak{b}$. Hence the additivity and cofinality of $\mathcal{N}^h_Ω$ can differ from those of the Lebesgue null ideal. We also show that the finite graphs in the Davies--Rogers construction can be chosen so that $\operatorname{cov}(\mathcal{N}^h_Ω)\le\operatorname{non}(\mathcal{N}^h_Ω)$.

math.LO↗