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Let T(Ω) be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT(Ω)0 is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA
0 + (Π
−TR).] In particular let ↦Ω
denote the enumeration function of the infinite...
We present some results on countable homogeneous 3-hypergraphs. In particular, we show that there is no unexpected homogeneous 3-hypergraph determined by a single constraint.
Assuming that an inaccessible cardinal exists, we construct a ZFC-model where every Δ
-set is measurable but there exists a Δ
-set without the property of Baire. By a result of Shelah, an inaccessible cardinal is necessary for this result.
We prove the “Main Gap” for the class ofa-models (sufficiently saturated models) of an arbitrary stable 1-based theoryT. We (i) prove a strong structure theorem fora-models, assuming NDOP, and (ii) roughly compute the number ofa-models ofT in any given cardinality.
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