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Constructing ω-stable structures: Computing rank

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Baldwin J., Holland K.

Fundamenta Mathematicae

2001

tom 170

nr 1-2

1-20

This is a sequel to [1]. Here we give careful attention to the difficulties of calculating Morley and U-rank of the infinite rank ω-stable theories constructed by variants of Hrushovski's methods. Sample result: For every k < ω, there is an ω-stable expansion of any algebraically closed field which has Morley rank ω × k. We include a corrected proof of the lemma in [1] establishing that the generic model is ω-saturated in the rank 2 case.

A Hanf number for saturation and omission

100%

Baldwin J., Shelah S.

Fundamenta Mathematicae

2011

tom 213

nr 3

255-270

Suppose t = (T,T₁,p) is a triple of two countable theories T ⊆ T₁ in vocabularies τ ⊂ τ₁ and a τ₁-type p over the empty set. We show that the Hanf number for the property 'there is a model M₁ of T₁ which omits p, but M₁ ↾ τ is saturated' is essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some distinctions between 'first order' and 'second order quantification'. In particular, we show that if κ is uncountable, then $h³(L_{ω,ω}(Q),κ) = h³(L_{ω₁,ω},κ)$, where h³ is the 'normal' notion of Hanf function (Definition 4.12).

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Baldwin J.

1 Holland K.

1 Shelah S.

1 2011

1 2001

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Constructing ω-stable structures: Computing rank

A Hanf number for saturation and omission