On the number of Dedekind cuts and two-cardinal models of dependent theories

Abstract

For an infinite cardinal $\kappa$, let $ded\kappa$ denote the supremum of the number of Dedekind cuts in linear orders of size $\kappa$. It is known that $\kappa <ded\kappa \le 2^{\kappa }$ for all $\kappa$ and that $ded\kappa <2^{\kappa }$ is consistent for any $\kappa$ of uncountable cofinality. We prove however that $2^{\kappa }\le ded(ded(ded(ded\kappa )))$ 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.

arXiv comment15 pages; some minor presentation improvements; Journal of the Institute of Mathematics of Jussieu, accepted

Cited by in corpus (1)

• On n-dependent groups and fields II