Abstract:In [5], Hjorth proved that for every countable ordinal α, there exists a complete Lω 1 ,ω -sentence φα that has models of all cardinalities less than or equal to ℵα, but no models of cardinality ℵ α+1 . Unfortunately, his solution does not yield a single Lω 1 ,ω -sentence φα, but a set of Lω 1 ,ω -sentences, one of which is guaranteed to work. It was conjectured in [9] that it is independent of the axioms of ZFC which of these sentences has the desired property.In the present paper, we prove that this conjectu… Show more
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