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

TL;DR

This paper investigates the relationship between Dedekind cuts and model sizes in dependent theories, establishing new bounds that influence the understanding of two-cardinal models and omitting types in such theories.

Contribution

It proves a new inequality relating Dedekind cuts and exponentials, and calculates Hanf numbers for two-cardinal models and type omission in dependent theories.

Findings

Proves that $2^{} \u2264 ded(ded(ded(ded)))$ for all .

Calculates Hanf numbers for two-cardinal models with large gaps.

Determines bounds for models omitting types in countable 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 $\kappa<ded\kappa\leq 2^{\kappa}$ for all $\kappa$ and that $ded\kappa<2^{\kappa}$ is consistent for any $\kappa$ of uncountable cofinality. We prove however that $2^{\kappa}\leq ded ( ded ( ded ( ded\kappa)))$ 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.