arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 7 new papers, 1 cross-list and 7 replacements in Logic (math.LO), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We study the equational theory and subvarieties of the variety $\mathcal V$ generated by $(\mathbb N_0,\min,+)$. We prove that a subvariety of $\mathcal V$ is proper if and only if it is locally finite, equivalently, if and only if it satisfies a power-stabilization identity. Every finite member generates a hereditarily finitely based variety with a finite subvariety lattice. In contrast, the periodic subvarieties defined by $x^m=x^{m+1}$ are nonfinitely based for $m\geq2$. We construct a strictly descending chain of nonfinitely based subvarieties and three varieties generated by explicit reduced products, each with a decidable equational theory. The smallest is a limit variety: it is nonfinitely based, and every proper subvariety has an absolute finite identity basis. The subvariety lattice of $\mathcal V$ has cardinality continuum. In fact, the interval between the first reduced-product variety and the subvariety defined by $x^2=x^3$ already has this cardinality, and all its members are nonfinitely based. We also give a hereditarily finitely based example with an infinite subvariety lattice. The constructions use finite convex models and recognition quotients.
Friedman and Stanley asked whether an invariant Borel class of countable structures must be Borel complete whenever every countable level of their hierarchy reduces to it. We construct a counterexample in one fixed countable language. The structural counterexample and its principal supporting results have been formalized in Lean~4 and it is available on github.
We introduce a combinatorial principle $\textsf{SC}(\kappa)$, for any regular cardinal $\kappa \ge \omega_2$, which asserts the existence of a family $\mathcal{Y} \subseteq [H(\kappa)]^\omega$ of elementary substructures of $H(\kappa)$ which is closed under finite intersections and satisfies that for stationarily many $N \in \mathcal{Y}$, if $M \in \mathcal{Y}$ and $M \cap \omega_1 < N \cap \omega_1$, then $M \cap N \cap \kappa \in N$. We explore the idea of defining forcings with side conditions consisting of finite subsets of such a family $\mathcal{Y}$. If $\textsf{SC}(\omega_2)$ holds, then there exists a thin stationary subset of $[\omega_2]^\omega$, and consequently $\textsf{SC}(\omega_2)$ is independent of ZFC. Assuming that $\mu$ is a Mahlo cardinal, the L\'{e}vy collapse $\mathrm{Col}(\omega,<\mu)$ forces that $\textsf{SC}(\kappa)$ holds for all regular cardinals $\kappa \ge \omega_2$. Local instances of $\textsf{SC}$ can be forced using matrices of models or derived as consequences of square principles. We apply the principle $\textsf{SC}$ to define two kinds of forcings: a streamlined version of a strongly proper collapse, and a forcing which is strongly proper on a stationary set and adds arbitrarily many Baumgartner clubs. Finally, we prove two consistency results using forcings derived from $\textsf{SC}$ related to Aronszajn trees, the guessing model principle, strong failures of club guessing, and the higher dominating number.
We prove that there is no sequence $\{ X_n \}_{n\in\omega}$ such that for all $n$, there is an injection from $\mathcal{P}(X_{n+1})$ to $X_n$ under a restricted set of axioms that conforms to the "working mathematician's" intuition (excluding Choice, Replacement, and Foundation). We also show such a sequence cannot be defined using a formula either. More precisely, it is inconsistent with Zermelo set theory minus Infinity to have a formula $\Phi(\cdot,\cdot)$ such that for each $n\in\omega$ there are $X$ and $Y$ with $\Phi(n,X)$, $\Phi(n+1,Y)$, and $\mathcal{P}(Y)$ being injectable into $X$. We finally show that the theorem fails if we allow infinitely many definitions, one for each term of the sequence, by using Hamkins, Linetsky, and Reitz's pointwise definable extensions of countable models of ZFC.
We study the relationship between large cardinals and Blass' Small Violation of Choice ($\mathsf{SVC}$), which states that the Axiom of Choice holds in a set-generic extension of the universe. Our main theorem is that in the presence of a supercompact cardinal, either the Axiom of Choice fails significantly, or else $\mathsf{SVC}$ holds in $L(\mathcal P(\mathrm{Ord}))$. We show that a supercompact is not strong enough to reflect $\mathsf{SVC}$, while an extendible cardinal is. If we assume Usuba's Lo\'s Property holds for the supercompact, then we can outright derive $\mathsf{SVC}$ in the universe.
We prove that the $\mathrm{P}$-ideal dichotomy implies the negation of the square principle $\square(\lambda, \omega)$ for every regular cardinal $\lambda\geq\omega_2$.
In [Pil07], Pillay showed that every imaginary in a proper pair of algebraically closed fields is interalgebraic with an imaginary of a certain algebraic form. In this paper, we establish an analogue for lovely pairs of pseudofinite fields. After naming countably many constants, we prove that every imaginary in lovely pairs of pseudofinite fields is interalgebraic with an imaginary of the same algebraic form as [Pil07] in the corresponding pairs of finite extensions of pseudofinite fields.
A space is ESLC if it contains neither an $S$-space nor an $L$-space, and NFS is the class of spaces containing no free sequences of length $\omega_1$. We prove that under $\mathrm{MA}(\omega_1)$, if all finite powers of $X$ belong to NFS, then $C_p(X)^\omega$ is ESLC; thus, for subspaces of $C_p(X)$, hereditary Lindel\"ofness, hereditary separability and countable spread coincide. This is derived from a dual form of a theorem of Todor\v{c}evi\'c. A Lindel\"of $\Sigma$-space belongs to NFS if and only if its compact subspaces have countable tightness. Hence $C_p(X)$ contains no $L$-spaces if $X$ is a Lindel\"of $\Sigma$-space of countable tightness and $\mathrm{MA}(\omega_1)$ holds, or if $X$ is any Lindel\"of $\Sigma$-space and PFA holds. Under $\mathrm{MA}(\omega_1)$, all iterated function spaces of Corson compact spaces are ESLC, and in ZFC the space $C_p(X,\{0,1\})$ is ESLC for every zero-dimensional Corson compact space $X$.