77 research outputs found
Set Theory
This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject
Maximal sets without Choice
We show that it is consistent relative to ZF, that there is no well-ordering
of while a wide class of special sets of reals such as Hamel
bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more
precise, we can assume that every projective hypergraph on has a
maximal independent set, among a few other things. For example, we get
transversals for all projective equivalence relations. Moreover, this is
possible while either holds, or countable choice for
reals fails. Assuming the consistency of an inaccessible cardinal, "projective"
can even be replaced with "". This vastly strengthens earlier
consistency results in the literature.Comment: 16 page
Set Theory
This meeting covered all important aspects of modern Set Theory, including large cardinal theory, combinatorial set theory, descriptive set theory, connections with algebra and analysis, forcing axioms and inner model theory. The presence of an unusually large number (19) of young researchers made the meeting especially dynamic
Equivalence Relations Which Are Borel Somewhere
The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I + Δ^1_1 sets ordered by ⊆ is a proper forcing. Let E be a Σ^1_1 or a ∏^1_1 equivalence relation on X with all equivalence classes Δ^1_1. If for all z ∈_H(2^N0)+, z ♯ exists, then there exists an I + Δ^1_1 set C ⊆ X such that E ↾ C is a Δ^1_1 equivalence relation
Stationary set preserving L-forcings and the extender algebra
Wir konstruieren das Jensensche L-Forcing und nutzen dieses um die Pi_2 Konsequenzen der Theorie ZFC+BMM+"das nichtstationäre Ideal auf omega_1 ist abschüssig" zu studieren. Viele natürliche Konsequenzen der Theorie ZFC+MM folgen schon aus dieser schwächeren Theorie. Wir geben eine neue Charakterisierung des Axioms Dagger ("Alle Forcings welche stationäre Teilmengen von omega_1 bewahren sind semiproper") in dem wir eine Klasse von L-Forcings isolieren deren Semiproperness äquivalent zu Dagger ist. Wir verallgemeinern ein Resultat von Todorcevic: wir zeigen, dass Rado's Conjecture Dagger impliziert. Des weiteren studieren wir Generizitätsiterationen im Kontext einer messbaren Woodinzahl. Mit diesem Werkzeug erhalten wir eine Verallgemeinerung des Woodinschen Sigma^2_1 Absolutheitstheorems. We review the construction of Jensen's L-forcing which we apply to study
the Pi_2 consequences of the theory ZFC + BMM + "the nonstationary
ideal on omega_1 is precipitous". Many natural consequences ZFC + MM
follow from this weaker theory. We give a new characterization of the
axiom dagger ("All stationary set preserving forcings are semiproper")
by isolating a class of stationary set preserving L-forcings whose
semiproperness is equivalent to dagger. This characterization is used to
generalize work of Todorcevic: we show that Rado's Conjecture implies
dagger. Furthermore we study genericity iterations beginning with a
measurable Woodin cardinal. We obtain a generalization of Woodin's
Sigma^2_1 absoluteness theorem
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