On a classification of theories without the independence property

Abstract

Key wordsThe independence property, stability, dependent theories. MSC (2010) 03C07, 03C45, 03C64A theory is stable up to Δ if any Δ-type over a model has a few extensions up to complete types. We prove that a theory has no the independence property iff it is stable up to some Δ, where each ϕ(x;ȳ) ∈ Δ has no the independence property.

Theories With Distal Shelah Expansions

“…The following concept will be useful in the proof of Lemma 1.3. The following is an immediate consequence of Fact 1 in [3] (see also [1,5]). We note that there is a minimum such C. Lemma 2.6.…”

“…) An ordinal (in this case a finite one) cannot be decreased infinitely many times. So, for each pair 1 , 2 , we can extend so that |C 1 B ∩ C 2 B | is minimised. We can enumerate all such pairs of formulas and extend appropriately for each one in turn, taking unions at limit ordinals.…”

A. D. Taimanov and model theory in Kazakhstan

“…Verbovskiy proved that dp-minimal theories with linear order are o-stable [339] (2010). Later he introduced the concept of a stable up to delta theory and proved that dependent theories are stable up to some subset of formulas without the independence property [341] (2013).…”

“…V.V. Verbovskiy introduced the notion of relative stability and proved a characterization of NIP theories via stability up to ∆ in [341] (2013). A theory is stable up to ∆ if any ∆-type over a model has few extensions up to complete types.…”