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ON THE NUMBER OF DEDEKIND CUTS AND TWO-CARDINAL MODELS OF DEPENDENT THEORIES
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 15 / Issue 4 / October 2016
- Published online by Cambridge University Press:
- 23 January 2015, pp. 771-784
- Print publication:
- October 2016
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- Article
- Export citation
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For an infinite cardinal
${\it\kappa}$, let 
$\text{ded}\,{\it\kappa}$ denote the supremum of the number of Dedekind cuts in linear orders of size 
${\it\kappa}$. It is known that 
${\it\kappa}<\text{ded}\,{\it\kappa}\leqslant 2^{{\it\kappa}}$ for all 
${\it\kappa}$ and that 
$\text{ded}\,{\it\kappa}<2^{{\it\kappa}}$ is consistent for any 
${\it\kappa}$ of uncountable cofinality. We prove however that 
$2^{{\it\kappa}}\leqslant \text{ded}(\text{ded}(\text{ded}(\text{ded}\,{\it\kappa})))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.
