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 , where h³ is the ’normal’ notion of Hanf function (Definition 4.12).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-3-5,
author = {John T. Baldwin and Saharon Shelah},
title = {A Hanf number for saturation and omission},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {255-270},
zbl = {1254.03072},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-3-5}
}
John T. Baldwin; Saharon Shelah. A Hanf number for saturation and omission. Fundamenta Mathematicae, Tome 215 (2011) pp. 255-270. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm213-3-5/