逻辑

Let $\mu$ be a strong limit singular cardinal. We prove that if $2^{\mu} > \mu^+$ then $\binom{\mu^+}{\mu}\to \binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ for every ordinal $\tau<\mu^+$. We obtain an optimal positive relation under $2^\mu = \mu^+$,…

We study Structural Reflection beyond Vop\v{e}nka's Principle, at the level of almost-huge cardinals and higher, up to rank-into-rank embeddings. We identify and classify new large cardinal notions in that region that correspond to some…

The Baire algebra of a topological space $X$ is the quotient of the algebra of all subsets of $X$ modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which…

Assuming an instance of the Brodsky-Rinot proxy principle holding at a regular uncountable cardinal $\kappa$, we construct $2^\kappa$-many pairwise non-embeddable minimal non-$\sigma$-scattered linear orders of size $\kappa$. In particular,…

In the paper hereditary classes of ${\rm L}$-structures are studied with language of the form ${{\rm L} = {\rm L_{fin}} \cup {\rm L_\infty}}$, where ${{\rm L_{fin}} = \langle R_1,R_2,\ldots, R_m, = \rangle}$ and ${{\rm L_\infty} = \langle…

We consider a certain class of infinitary rules of inference, called here restriction rules, using of which allows us to deduce complete theories of given models. The first instance of such rules was the $\omega$-rule introduced by Hilbert,…

It is proved that for every positive integer $n$, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph_n$ is at least $c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the Tukey class $\mathcal D_{\aleph_n} $ of…

We show that any positive characteristic tame Hahn field $\mathbb{F}((t^\Gamma))$ containing $t$ is decidable in $\mathcal{L}_t$, the language of valued fields with a constant symbol for $t$, if $\mathbb{F}$ and $\Gamma$ are decidable. In…

This paper analyzes infinitary nondeterministic computability theory. The main result is D $\ne$ ND $\cap$ coND where D is the class of sets decidable by infinite time Turing machines and ND is the class of sets recognizable by a…

\textit{Mereological fusion}, also known as \textit{composition} and \textit{sum}, was originally used by me as a primitive notion to axiomatize \textit{Extensional Mereology} wih \textit{atoms} in \cite{Ly22}. Here, I extend this idea to…

The construction of first-order logic and set theory gives rise to apparent circularities of mutual dependence, making it unclear which can act as a self-contained starting point in the foundation of mathematics. In this paper, we carry out…

This paper serves to define an extension, which we call dimensional Veblen, of Oswald Veblen's system of ordinal functions below the large Veblen ordinal. This is facilitated by iterating derivatives of ordinal functions along…

In this paper we use display calculus to show the decidability for normal modal logic K and some of its extensions.

We develop infinite-dimensional Ramsey theory for Fra\"iss\'e limits of finitely constrained free amalgamation classes in finite binary languages. We show that our approach is optimal and in particular, recovers the exact big Ramsey degrees…

We determine the strictly positive fragment $\mathsf{QPL}^+(\mathsf{HA})$ of the quantified provability logic $\mathsf{QPL}(\mathsf{HA})$ of Heyting Arithmetic. We show that $\mathsf{QPL}^+(\mathsf{HA})$ is decidable and that it coincides…

B\"uchi arithmetics $\mathop{\mathbf{BA}}\nolimits_n$, $n\ge 2$, are extensions of Presburger arithmetic with an unary functional symbol $V_n(x)$ denoting the largest power of $n$ that divides $x$. We explore the structure of non-standard…

Double Boolean algebras are algebras $\underline{D}=(D;\sqcap,\sqcup,\neg,\lrcorner,\bot,\top)$ of type $(2,2,1,1,0,0)$ introduced by Rudolf Wille to capture the equational theory of the algebra of protoconcepts. Every double Boolean…

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is…

We present a family of local involutive integral bounded residuated lattice-ordered commutative monoids (involutive residuated lattices, for short) having Boolean term, radical term (see \cite{CT12} and \cite{T23}), and satisfying GAP…

It is widely claimed that the natural axiom systems$\unicode{x2013}$including the large cardinal axioms$\unicode{x2013}$form a well-ordered hierarchy. Yet, as is well-known, it is possible to exhibit non-linearity and ill-foundedness by…