@MISC{Chernikov_definition, author = {Artem Chernikov}, title = {Definition}, year = {} }

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Abstract

Given a cardinal κ, we define ded κ = sup{|I | : I is a linear order with a dense subset of size ≤ κ}. ◮ In general the supremum may not be attained. ◮ The study of ded κ was initiated by Baumgartner [1]. ◮ In model theory it arises naturally when one wants to count types. Some equivalent ways to compute ded κ The following cardinalities are the same: 1. ded κ. 2. sup{λ: ∃ a linear order of size ≤ κ with λ Dedekind cuts} 3. sup{λ: ∃ a regular µ and a linear order of size ≤ κ with λ cuts of cofinality µ on both sides}. (by a theorem of Kramer, Shelah, Tent and Thomas) 4. sup{λ: ∃ a regular µ and a tree of size ≤ κ with λ branches of length µ} 5. sup{λ: ∃ a regular µ and a binary tree of size ≤ κ with λ branches of length µ} Basic properties of ded κ ◮ κ < ded κ ≤ 2 κ for every infinite κ. (for the first inequality, let µ be minimal such that 2 µ> κ, and consider the tree 2 <µ) ◮ ded ℵ0 = 2 ℵ0 ◮ Assume GCH, then ded κ = 2 κ for all κ. ◮ So is ded κ the same as 2 κ in general? Fact (Mitchell) For every κ with cf κ> ℵ0 it is consistent that ded κ < 2 κ.More properties ◮ (Kunen) If κ ℵ0 = κ then (ded κ)