We construct a model of $\mathsf{MA_{\aleph_1}}+\mathsf{OCA}_T$ where Baumgartner's Axiom fails, settling a question of Farah. Moreover, in the same model there is an $\aleph_1$-dense set of reals which is neither reversible nor increasing, answering a question of Marun, Shelah, and Switzer.
We show that a non-universal Polish group can induce a complete orbit equivalence relation, which answers a question of Sabok from \cite{OPENPROBLEMS}.
We give an example of an SOP theory $T$, such that any $L(M)$-formula $\varphi(x,y)$ with $|y|=1$ is NSOP. We show that any such $T$ must have the independence property. We also give a simplified proof of Lachlan's theorem that if every $L$-formula $\varphi(x,y)$ with $|x|=1$ is NSOP, then $T$ is NSOP.
We introduce an axiomatisation of when a model of the form $L(V_{κ+1})^M$ can be considered a ``$κ$-Solovay model''; we show a characterisation of $κ$-Solovay models; and we prove elementary equivalences between $κ$-Solovay models.
The aim of this paper is to study the class of quasicomplemented distributive nearlattices. We investigate $α$-filters and $α$-ideals in quasicomplemented distributive nearlattices and some results on ideals-congruence-kernels. Finally, we also study the notion of Stone distributive nearlattice and give a characterization by means $σ$-filters.
The main motivation of this paper is the study of first-order model theoretic properties of structures having their roots in modal logic. We will focus on the connections between ultrafilter extensions and ultrapowers. We show that certain structures (called bounded graphs) are elementary substructures of their ultrafilter extensions, moreover their modal logics coincide.
Let $\mathcal{E}$ denote the $σ$-ideal generated by closed null sets on the reals. We show that the uniformity and the covering of $\mathcal{E}$ can be added to Cichoń's maximum with distinct values. More specifically, it is consistent that $\aleph_1<\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{b}<\mathrm{non}(\mathcal{E})<\mathrm{non}(\mathcal{M})<\mathrm{cov}(\mathcal{M})<\mathrm{cov}(\mathcal{E})<\mathfrak{d}<\mathrm{non}(\mathcal{N})<\mathrm{cof}(\mathcal{N})<2^{\aleph_0}$ holds.
In this paper, we show how to construct for a given consistent theory $U$ a $Σ^0_1$-predicate that both satisfies the Löb Conditions and the Kreisel Condition ---even if $U$ is unsound. We do this in such a way that $U$ itself can verify satisfaction of an internal version of the Kreisel Condition.
We consider two generalizations of the recursion theorem, namely Visser's ADN theorem and Arslanov's completeness criterion, and we prove a joint generalization of these theorems.
We investigate the interaction between the product of invariant types and domination-equivalence. We present a theory where the latter is not a congruence with respect to the former, provide sufficient conditions for it to be, and study the resulting quotient when it is.
We develop the notion of coherent ultrafilters (extenders without normality or well-foundedness). We then use definable coherent ultraproducts to characterize any extension of a model $M$ in any fragment of $\mathbb{L}_{\infty, ω}$ that defines Skolem functions by a sufficiently complete (but in $ZFC$) coherent ultrafilter. We apply this method to various elementary classes and AECs.
We study an extension of \g propositional logic whose corresponding algebra is an ordered Abelian group. Then we expand the ideas to first-order case of this logic.
We combine two notions in AECs, tameness and good $λ$-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.
We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,...,ω,\continuum}$ under which there exists a continuous function $F:[0,1]\to [0,1]$ such that for every $y\in[0,1]$ we have $|F^{-1}(y)|=f(y)$.
Under some cardinal arithmetic assumptions, we prove that every stationary subset of lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond- colouring restrictions. We then deal with semi- saturated (normal) filters.