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arXiv · 1308.3099

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

Artem Chernikov·Saharon Shelah
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Abstract

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

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Artem Chernikov, Saharon Shelah. 2014-12-30. On the number of Dedekind cuts and two-cardinal models of dependent theories. https://doi.org/10.1017/s1474748015000018

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