A strengthening of the MCFL-ness of $O_2$

Even smaller universal posets

We show that for every $\eta>0$ and sufficiently large $n$, there exists a poset of size $2^{(1+\eta)n/2}$ containing all the $n$-element posets as induced subposets. This improves a recent result of Bastide, Groenland and Nenadov. Our proof provides a labeling scheme preserving transitivity, inspired by the Boolean lattice. Among other tools, we use the Szemer\'edi Regularity Lemma.

Splendid extensions

Let $\kappa$ be a successor cardinal. We force a universe in which every model of PA of size $\kappa$ extends to a model M of the same size, where M has no splendid extensions. If there is an ineffable cardinal then this statement holds at some cardinal below it, in ZFC.

Failure of Higher-Order Truth within Intuitionistic Propositional Logic

We answer the question whether all Heyting algebras can appear as the lattice of subterminal objects of an elementary topos in the negative. Concretely, we have shown that the free Heyting algebra on two generators cannot be such a Heyting algebra. The mathematical results in this document were obtained with the help of ChatGPT 5.6 Sol, although the document itself was written entirely by us and we take full responsibility for its contents.

Proof reflection in Coq

A/-TERM REDUCTIONS (

Positive Lower Density for Hofstadter's $ab-1$ Problem

Let $A$ be the smallest set of positive integers containing $2$ and $3$ such that $ab-1\in A$ whenever $a,b\in A$ are distinct. We prove that $A$ has positive lower density, answering a problem of Erd\H{o}s attributed to Hofstadter.