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Theories without the tree property of the second kind

Artem Chernikov

arXiv:1204.0832v3math.LO

TL;DR

The paper studies NTP2 as a broader setting for methods associated with simplicity and NIP, developing results about burden, types, and forking. It proves structural and valued-field results, including an NTP2 criterion for Henselian valued fields of characteristic (0,0).

  • Problem

    NTP2 theories include important simple and NIP cases, but may have neither, motivating a systematic study of their structure and methods.

    Source: Introduction · pp. 1–2

  • Method

    The paper develops inp-pattern and burden arguments and studies forking, simple and NIP types, valued fields, and generic-predicate expansions.

    Source: Introduction · pp. 2–3

  • Results

    Burden is sub-multiplicative in arbitrary theories, and a Henselian valued field of characteristic (0,0) is NTP2 exactly when its residue field is NTP2.

    Source: Introduction · pp. 2–3

  • Takeaways & Limitations

    Every ultraproduct of p-adics is NTP2, and adding a generic predicate to a geometric NTP2 theory preserves NTP2.

    Source: Introduction · p. 3

    The generic-predicate question is left open without exchange for algebraic closure, including whether κinp and strongness are preserved.

    Source: §7.1 · p. 30

Abstract

from arXiv · show

We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2 then there is a formula with a single variable witnessing this); NTP2 is equivalent to the generalized Kim's lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters - so the dp-rank of a 1-type in any theory is always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic (0,0) is NTP2 (strong, of finite burden) if and only if the residue field is NTP2 (the residue field and the value group are strong, of finite burden respectively), so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2.

Introduction

The paper develops NTP2 as a setting for extending methods from simplicity and NIP, while studying simple and NIP types as important special cases. Its results concern burden, forking, valued fields, and generic predicates.

Section 2

  • Burden is sub-multiplicative in arbitrary theories, and any theory with TP2 has a formula with a singleton variable witnessing TP2.The value of κinp does not depend on the number of variables used to compute it; whether burden is sub-additive in NTP2 theories remains open.Source: Introduction · p. 2

Section 4

  • NTP2 is characterized through variants of Kim’s lemma, local character, and bounded weight relative to strict non-forking; dependent dividing implies NTP2.The analysis of also extends to work inside an NTP2 type in an arbitrary theory.Source: Introduction · p. 2

Introduction

  • The paper frames NTP2 through the principle “NTP2 = simplicity + NIP,” while noting that NTP2 theories may have neither simple nor NIP types.The class contains both simple and NIP theories and drew renewed interest alongside efforts to extend stability-theoretic methods to broader settings.Source: Introduction · pp. 1–2

Section 5

  • For NIP types, dp-rank in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations after adding parameters.In NTP2 theories, a type is NIP exactly when every extension has boundedly many global non-forking extensions.Source: Introduction · p. 2

Section 6

  • In NTP2 theories, simple types are co-simple, satisfy the co-independence theorem characterization, and have full forking symmetry with arbitrary elements.The paper also shows co-simple stably embedded types are simple.Source: Introduction · p. 3

Section 7

  • A Henselian valued field of characteristic (0,0) is NTP2 exactly when its residue field is NTP2; in particular, ultraproducts of p-adics are NTP2.The paper also reports the corresponding strongness and finite-burden conditions, and that adding a generic predicate to a geometric NTP2 theory preserves NTP2.Source: Introduction · p. 3

1. Preliminaries

The preliminaries establish terminology for indiscernible sequences and arrays and record tools for extracting mutually indiscernible arrays.

Preliminaries

  • Sequences are mutually indiscernible over C when each sequence is indiscernible over C together with all the other sequences.An array is also called mutually indiscernible when its rows are mutually indiscernible.Source: §1 · p. 3
  • From a sufficiently large finite-row array, one can extract an infinite-row-length array with mutually indiscernible rows while preserving the types of finite subarrays over C.The extraction applies when the entries have size at most κ, for a suitable array length λ.Source: §1 · p. 4
  • An invariant global type yields a sequence indiscernible over any larger set D when each element realizes the type restricted to D and its predecessors.The sequence’s type over D does not depend on the choice of its elements.Source: §1 · pp. 4–5

2. Burden and κinp

This section defines burden through inp-patterns and proves a product bound that yields sub-multiplicativity. It also records sub-additivity as an open question.

Burden and κinp

  • Burden is the supremum of the depths of inp-patterns in a type, and it is zero exactly for algebraic types.The section defines bdn(a/C) as the burden of tp(a/C).Source: §2 · p. 5
  • An inp-pattern of depth κ in a type yields a product-depth dichotomy: a pattern of depth κ1 × κ2 in tp(b1b2/C) forces depth κ1 in tp(b1/C) or depth κ2 in tp(b2/b1C).This is the theorem underlying the section’s sub-multiplicativity result for burden.Source: §2 · pp. 6–7
  • Sub-additivity of burden remains open both for NTP2 theories and for arbitrary theories.The text notes sub-additivity in NIP and simple theories before posing the broader questions.Source: §2 · p. 7

3. NTP2 and its place in the classification hierarchy

The section locates NTP2 among familiar dividing lines and compares burden with dp-rank and weight. It then examines the SOPn hierarchy under the NTP2 restriction.

3. NTP2 and its place in the classification hierarchy

  • If a theory has TP2, some formula with a singleton x-variable has TP2; NIP formulas are NTP2, and simple theories are NTP2.The section also states that disjunction preserves NTP2, whereas the triangle-free random graph shows conjunction need not preserve it.Source: §3 · pp. 8–10
  • For any partial type, burden is at most dp-rank; for an NIP type, the two ranks coincide.The section defines dp-rank using mutually indiscernible sequences that cease to be indiscernible over a realization.Source: §3 · pp. 8–9
  • The triangle-free random graph has TP2, and its formula xRy is NTP2, showing that a conjunction of NTP2 formulas need not be NTP2.Source: §3 · pp. 9–10

3.1. On the SOPn hierarchy restricted to NTP2 theories.

  • Within NTP2 theories, the final two items in the stated SOPn implication chain coincide.Source: §3.1 · p. 10
  • Examples with no directed cycles of length ≤n have SOPn but not SOPn+1; all these examples have TP2.Source: §3.1 · p. 10
  • The paper asks whether the SOPn hierarchy remains strict when restricted to NTP2 theories.The standard examples supplied for strictness have TP2, so they do not settle that question.Source: §3.1 · pp. 10–11

4. Forking in NTP2

This section develops NTP2 analogues of simplicity’s forking and dividing criteria, characterizing NTP2 through bounded ist-weight, Kim’s lemma, and generic local character. It also examines strongness and local character, relates dependent dividing to NTP2 and simplicity, and develops forking and dividing results for NTP2 types.

4. Forking in NTP2

  • A global type invariant over A is strictly invariant over any |A|+-saturated M containing A, and strict invariance supports making indiscernible sequences mutually C-indiscernible.The sequence construction preserves the types over their initial elements.Source: §4 · p. 12
  • In NTP2 theories, types over models have global strictly invariant extensions, and dividing and forking coincide over models and, for nonforking versus nondividing, over extension bases.These forking and dividing facts are recalled from.Source: §4 · p. 12

4.1. Characterization of NTP2.

  • An inp-pattern of depth κ in a partial type is equivalent to a realization with an ist-independent sequence whose members divide over an enlarged parameter set.The sequence and parameter set extend the partial type’s original base.Source: §4.1 · pp. 12–13
  • NTP2 is equivalent to absolutely bounded or bounded ist-weight, Kim’s lemma, and generic local character for dividing.The theorem states these conditions are equivalent to each other and to T being NTP2.Source: §4.1 · p. 13
  • The proof also characterizes strongness through finite or singleton witnesses and finite local-character extensions.The stated equivalences include strongness and conditions quantified over finite or singleton tuples, with a finite subset A′ used for types over B.Source: §4.1 · p. 15
  • For NTP2 theories, a small parameter set can be enlarged so that instances in a type whose parameters do not fork over the enlargement do not divide over it.Proposition 4.12 gives this local-character property for a type over a saturated model and an initial set of size below the model's saturation cardinal.Source: §4.1 · p. 15

4.2. Dependent dividing.

  • If dividing is witnessed by dependent formulas, the theory is NTP2; if dividing is witnessed by simple formulas, the theory is simple.Source: §4.2 · p. 15

4.3. Forking and dividing inside an NTP2 type.

  • In an NTP2 type, dividing of the type together with a formula yields a global coheir whose sequence of corresponding instances makes the type inconsistent.This localizes a forking-and-dividing tool to the type even without assuming that the ambient theory is NTP2.Source: §4.3 · p. 16
  • For an NTP2 type, a strictly invariant extension of a parameter type makes the corresponding sequence of divided instances inconsistent.The result assumes the type together with the initial formula divides over the base and applies to sequences chosen by successive restrictions of that invariant extension.Source: §4.3 · p. 16
  • If a formula is consistent with an NTP2 type without dividing over the base, it extends to a complete type over the type's realizations and the base that does not divide over the base.Source: §4.3 · p. 17

5. NIP types

The section studies NIP types, showing how dp-rank witnesses can be drawn from a type’s realizations even in arbitrary theories. It also characterizes NIP types inside NTP2 theories by boundedness of global non-forking extensions.

5. NIP types

  • A partial type is NIP exactly when its dp-rank is finite.Source: §5 · p. 17

5.1. Dp-rank of a type is always witnessed by an array of its realizations.

  • For an NIP partial type with dp-rank at least κ, mutually indiscernible sequences of its realizations witness that rank after adding parameters.The sequences are not indiscernible over the witnessing realization together with the enlarged parameter set.Source: §5.1 · p. 18
  • Consequently, the dp-rank of any 1-type is witnessed by mutually indiscernible sequences of singletons.Source: §5.1 · p. 18
  • For dependent partial types, burden can be witnessed by mutually indiscernible sequences of realizations; the analogous burden question has positive answers when T has dependent forking or is simple.Source: §5.1 · pp. 18–19

5.2. NIP types inside an NTP2 theory.

  • In an NTP2 theory, a partial type is NIP exactly when every extension has boundedly many global non-forking extensions.Source: §5.2 · p. 20

6. Simple types

The section characterizes simple and co-simple types through dividing, local character, rank, and co-independence conditions, and examines when co-simplicity implies simplicity. It also establishes that in NTP2 theories simple types are co-simple and have strong symmetry and independence-theorem properties, while giving examples and conditions that distinguish the two notions.

6.1. Simple and co-simple types.

  • Simplicity is equivalent to local character and Kim’s lemma, and to finiteness of D(p, ∆, k) for every finite ∆ and k; extensions of simple types remain simple.For simple types over extension bases, forking and dividing also coincide.Source: §6.1 · pp. 21–22
  • Co-simplicity is defined by dual tree-property, local-character, and sequence conditions, and a co-simple type can fail to be simple.The paper’s triangle-free random graph example gives a co-simple non-simple type over a model.Source: §6.1 · pp. 22–23

6.2. Simple types are co-simple in NTP2 theories.

  • For a simple type over an extension base in an NTP2 theory, the weak chain condition and mutual-indiscernibility construction provide key tools for establishing these results.The weak chain condition produces an indiscernible sequence independent from a realization; another lemma replaces Morley sequences in the type with mutually indiscernible sequences.Source: §6.2 · pp. 23–24
  • In NTP2 theories, dividing over an extension base for a formula with a simple-type realization has a witnessing Morley sequence, and forking between such realizations and arbitrary elements is symmetric.These results support the section’s conclusion that simple types are co-simple in NTP2 theories.Source: §6.2 · p. 25
  • Every simple type in an NTP2 theory is co-simple, and its nonforking relation has right transitivity.Source: §6.2 · pp. 25–26

6.3. Independence and co-independence theorems.

  • The paper gives localized and dual versions of Kim’s result on failure of the independence theorem when TP1 holds, without requiring a large cardinal assumption [Kim01].Source: §6.3 · p. 26
  • For NTP2 partial types, satisfying the independence theorem implies co-simplicity, while co-independence for every model extension implies simplicity.Source: §6.3 · p. 26
  • For a simple type over a model in an NTP2 theory, the independence theorem holds: compatible nonforking realizations have a common nonforking amalgam.Source: §6.3 · p. 27
  • In NTP2 theories, a partial type is simple exactly when every extension to a model satisfies the co-independence theorem, equivalently when forking with its realizations is symmetric.Source: §6.3 · p. 27
  • In NTP2, a stably embedded co-simple type is simple; when the type is a definable set, the argument works in arbitrary theories.Source: §6.3 · p. 28

7. Examples

The examples include an NTP2 result for adding a random graph to an o-minimal theory and inheritance results for Henselian valued fields. The valued-field section also describes indiscernible-sequence behavior and consequences for Hahn series and ultraproducts of p-adics.

7. Examples

  • Section 7 begins with a general lemma intended to simplify checking NTP2 in examples.Source: §7 · p. 28

7.1. Adding a generic predicate.

  • Adding a directed random graph to an o-minimal theory yields an NTP2 theory.Source: §7.1 · p. 30
  • Whether the generic-predicate result holds without exchange for algebraic closure, and whether κinp or strongness is preserved, is left as a problem.Source: §7.1 · p. 30

7.2. Valued fields.

  • If the residue field k is NTP2, then the Henselian valued field K is NTP2.If k and the value group Γ are strong, then K is strong; if both have finite burden, then K has finite burden.Source: §7.2 · p. 30
  • Hahn series over pseudo-finite fields are NTP2.Source: §7.2 · p. 30
  • An ultraproduct of p-adics is strong of finite burden in the Denef-Pas language, with burden at least 2.The burden lower bound is witnessed by the angular-component and valuation formulas.Source: §7.2 · p. 30
  • For an indiscernible sequence, the values v(c_j − c_i) for i < j are strictly increasing in i, strictly decreasing in j, or constant.The increasing case is called pseudo-convergent.Source: §7.2 · p. 31
  • For a pseudo-convergent indiscernible sequence and any a, a cut h determines how v(a − c_i) and ac(a − c_i) relate to the corresponding values for a limit element c_∞.On either side of h, the valuation and angular component equalities follow the distinct patterns stated in the passage.Source: §7.2 · p. 31
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