5,250 research outputs found
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
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