[Paper Review] Classification theory for theories with NIP - a modest beginning

Saharon Shelah|ArXiv.org|Sep 6, 2000

Advanced Topology and Set Theory3 citations

TL;DR

This paper initiates a classification theory for first-order theories with NIP (no independence property), extending model-theoretic tools like indiscernible sequences, averages, and perpendicularity to unstable yet NIP contexts. It establishes foundational results on definable types, duality of cofinality, and introduces invariants such as dual-cf and w-invariants, laying groundwork for a structural theory akin to stability and simplicity.

ABSTRACT

A relevant thesis is that for the family of complete first order theories with NIP (i.e. without the independence property) there is a substantial theory, like the family of stable (and the family of simple) first order theories. We examine some properties.

Motivation & Objective

• To develop a classification theory for NIP first-order theories analogous to the well-established theories of stable and simple theories.
• To investigate the behavior of indiscernible sequences and their averages in NIP contexts, especially when the theory is unstable.
• To define and analyze invariants such as dual-cf and w-invariants for endless indiscernible sequences to capture structural complexity.
• To explore the role of definable types, canonical bases, and quasi-orders in NIP theories, particularly in relation to the order property and infinite chains.
• To lay the groundwork for future generalizations, such as super-NIP or dimension-like invariants, by identifying test questions and open problems in the framework.

Proposed method

• Uses ultrafilters and types to define the average of a sequence (Av(D, J)), enabling analysis of definable behavior in NIP models.
• Introduces the concept of D-indiscernible sequences over a set A, where type-definability is governed by ultrafilters and average types.
• Defines key classes of formulas: avf (averagable), daf (definable as finite union of convex sets), and daf^n (bounded convexity), to classify formula behavior over sequences.
• Applies the dual-cf invariant to measure the complexity of indiscernible sequences, showing that dual-cf(𝕀, M) = θ₂ leads to contradiction under certain saturation assumptions.
• Employs model-theoretic saturation arguments and absoluteness to derive structural constraints, particularly in the context of large cardinals and cofinality.
• Uses combinatorial approximations (e.g., θ₂-approximations) and symmetric relations R ⊆ λ × λ to define and analyze the dual-cf-κ-dimensional independence property.

Experimental results

Research questions

• RQ1 Can a classification theory for NIP theories be developed that parallels the theories of stable and simple theories?
• RQ2 What structural properties do indiscernible sequences in NIP theories exhibit, especially when the theory is unstable?
• RQ3 How do invariants like dual-cf(𝕀, M) and w(𝕀) characterize the complexity and orthogonality of endless indiscernible sequences?
• RQ4 To what extent can definable types and canonical bases be generalized in NIP theories, particularly in the absence of stability?
• RQ5 Can the dual-cf-κ-dimensional independence property be used to distinguish or characterize NIP theories, especially in relation to p-adics or other known examples?

Key findings

• For any NIP theory T, every formula φ(x, y) divides any indiscernible sequence into finitely many convex sets, generalizing a key feature of stable theories.
• The dual-cf(𝕀, M) invariant measures the complexity of an endless indiscernible sequence 𝕀 over a model M, and if dual-cf(𝕀, M) = θ₂, it leads to a contradiction under saturation assumptions.
• A dichotomy holds: a type p ∈ Sm(M) is stable if and only if it is definable and every ultrafilter realizing it generates an indiscernible set; otherwise, it supports an infinite chain in a quasi-order.
• If T is unstable but NIP, then some formula φ(x, y; c) defines a quasi-order with infinite chains, and such formulas may also have the order property (though not necessarily Eherenfeucht’s property E).
• The paper shows that the construction in §5 yields models with more specific freedom than instability alone would suggest, suggesting richer internal structure in NIP theories.
• The paper introduces the w(𝕀) invariant as the supremum of lengths of chains of pairwise non-perpendicular endless indiscernible sequences, though it does not behave like algebraic dimension.