Submission date: 14 August 2013.

Abstract:

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

Mathematics Subject Classification: 03E04, 03E10, 03E75, 03C45, 03C55

Keywords and phrases:

Full text arXiv 1308.3099: pdf, ps.