The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_σ$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof extends to larger classes of sets.
We use indecomposable ultrafilters to answer some questions of Hayut, Karagila paper "Spectra of uniformity". It is shown that the bound on the strength by T. Usuba "A note on uniform ultrafilters in choiceless context" is optimal.
In the lecture notes it is shown that an ordinal $ψ_Ω(\varepsilon_{\mathbb{S}^{+}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory ${\sf KP}ω+(M\prec_{Σ_{1}}V)$. In this note we show that ${\sf KP}ω+(M\prec_{Σ_{1}}V)$ proves the well-foundedness up to $ψ_Ω(ω_{n}(\mathbb{S}^{+}+1))$ for each $n$.
We construct a model in which MA$_{ω_1}$(S)[S] holds and $\mathcal{K}_2$ fails. This shows that MA$_{ω_1}$(S)[S] does not imply $\mathcal{K}_2$ and answers an old question of Larson and Todorcevic in [3]. We also investigate different strong colorings in models of MA$_{ω_1}$(S)[S].
We prove that it is consistent with large values of the continuum that there are no S-spaces. We also show that we can also have that compact separable spaces of countable tightness have cardinality at most the continuum.
We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not $σ$-continuous with closed witnesses.
We prove that uniform metastability is equivalent to all closed subspaces being pseudocompact and use this to provide a topological proof of the metatheorem introduced by Caicedo, Duenez and Iovino on uniform metastability and countable compactness for logics.
We present the notions of positively complete theory and general forms of amalgamation in the framework of positive logic. We explore the fundamental properties of positively complete theories and study the behaviour of companion theories by a change of constants in the language. Moreover, we present a general form of amalgamation and discuss some forms of strong amalgamation.
Gödel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing that the contradiction that emerges at the end should be redirected to refute this initial assumption rather than the completeness of number theory
El pasado 27 de abril tuvo lugar en MediaLab Prado Madrid la conferencia “Lo que pasa y lo que queda” a cargo de Iñaki Gabilondo. Este acto se produjo en el marco del 15º aniversario de los Estudios de Ciencias de la Información y de la Comunicación de la UOC (#UOC15infocom). El ponente apuntó interesantes reflexiones sobre el futuro de la información y la comunicación en un mundo en radical transformación.
We complete the characterization of the possible spectrum of regular ultrafilters D on a set I, where the spectrum is the set of infinite cardinals which are ultraproducts of finite cardinals modulo D.
We prove, using a weakening of the Proper Forcing Axiom, that any homemomorphism between Cech--Stone remainders of any two locally compact, zero-dimensional Polish spaces is induced by a homeomorphism between their cocompact subspaces.
We survey the theory of Muchnik (weak) and Medvedev (strong) degrees of subsets of ${}^ωω$ with particular attention to the degrees of $Π^0_1$ subsets of ${}^\omega2$. Later sections present proofs, some more complete than others, of the major results of the subject.
Let $\bfΓ$ be a Borel class, or a Wadge class of Borel sets, and $2\leq d\leqω$ a cardinal. We study the Borel subsets of ${\mathbb R}^d$ that can be made $\bfΓ$ by refining the Polish topology on the real line. These sets are called potentially $\bfΓ$. We give a test to recognize potentially $\bfΓ$ sets.
We present a short proof of Szemerédi's Theorem using a dynamical system enriched by ideas from model theory. The resulting proof contains features reminiscent of proofs based on both ergodic theory and on hypergraph regularity.
We give sufficient conditions for a predicate P in a complete theory T to be stably embedded: P with its induced 0-definable structure has "finite rank", P has NIP in T and P is 1-stably embedded. This generalizes recent work by Hasson and Onshuus in the case where P is o-minimal in T.