Lebesgue's Density Theorem and definable selectors for ideals

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio

Éducation à l‘environnement et efficacité : un outil de travail

Au fil des ans, l’éducation à l’environnement a acquis une grande expérience pratique et la qualité des offres est élevée. Avec la diminution des ressources, l’accent est désormais mis davantage sur l’efficacité. La brochure « Éducation à l‘environnement et efficacité » est conçue pour vous aider à identifier, améliorer et mesurer l’efficacité de vos offres de formation

Umweltbildung mit Wirkung : eine Arbeitshilfe

Umweltbildung hat über die Jahre viel Praxiserfahrung aufgebaut, die Qualität der Angebote ist hoch. Mit knapper werdenden Ressourcen rückt nun vermehrt die Wirkung der Umweltbildung ins Blickfeld. Mit einer Arbeitshilfe werden Anbietende dabei unterstützt, die Wirkung Ihrer Angebote benennen zu können, zu steigern und zu messen

The consistency strength of long projective determinacy

We determine the consistency strength of determinacy for projective games of length omega(2). Our main theorem is that Pi(1)(n+1)-determinacy for games of length omega(2) implies the existence of a model of set theory with omega + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M-n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A = R and the Axiom of Determinacy. Then we argue how to obtain a model with omega + n Woodin cardinal from this. We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length omega(2) with payoff in (sic)(R)Pi(1)(1) or with sigma-projective payoff