A Hanf number for saturation and omission

John T. Baldwin, and Saharon Shelah. "A Hanf number for saturation and omission." Fundamenta Mathematicae 213.3 (2011): 255-270. <http://eudml.org/doc/286236>.

@article{JohnT2011,
abstract = {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).},
author = {John T. Baldwin, Saharon Shelah},
journal = {Fundamenta Mathematicae},
keywords = {Hanf number; second-order logic; saturated models; omitting types},
language = {eng},
number = {3},
pages = {255-270},
title = {A Hanf number for saturation and omission},
url = {http://eudml.org/doc/286236},
volume = {213},
year = {2011},
}

TY - JOUR
AU - John T. Baldwin
AU - Saharon Shelah
TI - A Hanf number for saturation and omission
JO - Fundamenta Mathematicae
PY - 2011
VL - 213
IS - 3
SP - 255
EP - 270
AB - 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).
LA - eng
KW - Hanf number; second-order logic; saturated models; omitting types
UR - http://eudml.org/doc/286236
ER -