We will give a two-sort system which axiomatizes winning strategies for the combinatorial game Node Kayles. It is shown that our system captures alternating polynomial time reasonings in the sense that the provably total functions of the theory corresponds to those computable in APTIME. We will also show that our system is equivalently axiomatized by Sprague–Grundy theorem which states that any Node Kayles position is provably equivalent to some NIM heap.

We prove a compactness theorem for pseudopower operations of the form $${{\,\mathrm{pp}\,}}_{\Gamma (\mu ,\sigma )}(\mu )$$ where $$\aleph _0<\sigma ={{\,\mathrm{cf}\,}}(\sigma )\le {{\,\mathrm{cf}\,}}(\mu )$$ . Our main tool is a result that has Shelah’s cov versus pp Theorem as a consequence. We also show that the failure of compactness in other situations has significant consequences for pcf theory, in particular, implying the existence of a progressive set A of regular cardinals for which $${{\,\mathrm{pcf}\,}}(A)$$ has an inaccessible accumulation point.

Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is $$\aleph _2$$ . The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from the Proper Forcing Axiom, for example, the P-ideal Dichotomy. In this article, it is proved that the Y-Proper Forcing Axiom implies the Mapping Reflection Principle by introducing forcing notions whose conditions are finite objects.

Continuing (Fuchino et al. in Arch Math Log, 2020. https://doi.org/10.1007/s00153-020-00730-x), we study the Strong Downward Lowenheim–Skolem Theorems (SDLSs) of the stationary logic and their variations. In Fuchino et al. (2020) it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters $$\textsf {SDLS}({\mathcal {L}}^{\aleph _0}_{stat},{<}\,\aleph _2)$$ down to $${<}\,\aleph _2$$ is equivalent to the conjunction of CH and Cox’s Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters $$\textsf {SDLS}^-({\mathcal {L}}^{\aleph _0}_{stat},{<}\,2^{\aleph _0})$$ down to $${<}\,2^{\aleph _0}$$ implies that the size of the continuum is $$\aleph _2$$ . In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to $${<}\,2^{\aleph _0}$$ under the continuum being of size $$>\aleph _2$$ . This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size $${<}\,2^{\aleph _0}$$ . We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation $$\textsf {SDLS}^{int}_+({\mathcal {L}}^{PKL}_{stat},{<}\,2^{\aleph _0})$$ for reflection down to $${<}\,2^{\aleph _0}$$ is consistent under the assumption of the consistency of ZFC $$+$$ “the existence of a supercompact cardinal” and this SDLS implies that the continuum is (at least) weakly Mahlo. These three “axioms” in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Laver-generic supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be $$\aleph _1$$ or $$\aleph _2$$ or very large respectively. We also show that the existence of one of these generic large cardinals implies the “ $$++$$ ” version of the corresponding forcing axiom.

We show that for every sofic chunk $E$ there is a bijective homomorphism $f: E_c \rightarrow E$, where $E_c$ is a chunk of the group of computable permutations of $\mathbb{N}$ so that the approximating morphisms of $E$ can be viewed as restrictions of permutations of $E_c$ to finite subsets of $\mathbb{N}$. Using this we study some relevant effectivity conditions associated with sofic chunks and their profiles.

We prove that every usco multimap varPhi :Xrightarrow Y from a metrizable separable space X to a GO-space Y has an F_sigma -measurable selection. On the other hand, for the split interval {ddot{mathbb I}} and the projection P:{{ddot{mathbb I}}}^2rightarrow mathbb I^2 of its square onto the unit square mathbb I^2, the usco multimap {P^{-1}:mathbb I^2multimap {{ddot{mathbb I}}}^2} has a Borel (F_sigma -measurable) selection if and only if the Continuum Hypothesis holds. This CH-example shows that know results on Borel selections of usco maps into fragmentable compact spaces cannot be extended to a wider class of compact spaces.

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.

We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{\omega_1,\omega}$-sentence $\psi$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $\psi$ is consistently equal to $[\aleph_0,\aleph_{\omega_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $\psi$ is consistently equal to $[\aleph_1,\aleph_{\omega_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathcal{L}_{\omega_1,\omega}$-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in [18]. (3) Consistently, $\psi$ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal models in countably many cardinalities. (4) $2^{\aleph_0}<\aleph_{\omega_1}<2^{\aleph_1}$ and there exists an $\mathcal{L}_{\omega_1,\omega}$-sentence with models in $\aleph_{\omega_1}$, but no models in $2^{\aleph_1}$. This relates to a conjecture by Shelah that if $\aleph_{\omega_1}<2^{\aleph_0}$, then any $\mathcal{L}_{\omega_1,\omega}$-sentence with a model of size $\aleph_{\omega_1}$ also has a model of size $2^{\aleph_0}$. Our result proves that $2^{\aleph_0}$ can not be replaced by $2^{\aleph_1}$, even if $2^{\aleph_0}<\aleph_{\omega_1}$.

We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \(\mathsf {cov}(\mathcal {M})\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.

Tarski algebras, also known as implication algebras or semi-boolean algebras, are the $$\left\{ \rightarrow \right\} $$ -subreducts of Boolean algebras. In this paper we shall introduce and study the complete and atomic Tarski algebras. We shall prove a duality between the complete and atomic Tarski algebras and the class of covering Tarski sets, i.e., structures $$\left $$ , where X is a non-empty set and $${\mathcal {K}}$$ is non-empty family of subsets of X such that $$\bigcup {\mathcal {K}}=X$$ . This duality is a generalization of the known duality between sets and complete and atomic Boolean algebras. We shall also analize the case of complete and atomic Tarski algebras endowed with a complete modal operator, and we will prove a duality for these algebras.