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A New Kim's Lemma

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Alex Kruckman, Nicholas Ramsey

Logic · Submitted 6 Sep 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP2 and NSOP1 variants.

how it is stated · asserted 1.00 read the record →

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arXiv ID2309.02718 (version 2)
Pith IDLAX3PSQZ, Pith's permanent record of this paper and its review
CitationAlex Kruckman, Nicholas Ramsey. "A New Kim's Lemma." arXiv:2309.02718 (2023). https://pith.science/paper/LAX3PSQZ
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,206 input tokens.

What the abstract itself states, as probabilities: claims first 0.13 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.86 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Kim's Lemma is a key ingredient in the theory of forking independence in simple theories. It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters. Variants of Kim's Lemma have formed the core of the theories of independence in two orthogonal generalizations of simplicity - namely, the classes of NTP2 and NSOP1 theories. We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP2 and NSOP1 variants. We explore examples and non-examples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Kim's Lemma is a key ingredient in the theory of forking independence in simple theories. background 1.00 0.04 0.02 asserted 1.00
s1 It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters. background 1.00 0.04 0.03 asserted 1.00
s2 Variants of Kim's Lemma have formed the core of the theories of independence in two orthogonal generalizations of simplicity - namely, the classes of NTP2 and NSOP1 theories. background 1.00 0.05 0.04 asserted 1.00
s3 We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP2 and NSOP1 variants. contribution 0.99 0.94 0.81 asserted 1.00
s4 We explore examples and non-examples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions. contribution 0.93 0.93 0.13 asserted 0.80

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Kim's Lemma is a key ingredient in the theory of forking independence in simple theories. It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters. Variants of Kim's Lemma have formed the core of the theories of independence in two orthogonal generalizations of simplicity - namely, the classes of NTP2 and NSOP1 theories. We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP2 and NSOP1 variants. We explore examples and non-examples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions.

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How the literature uses it

1 arXiv paper cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 5 sentences without a confident label

Compares with 3 papers
  • In the context of $$, they also study dividing along coheir sequences which are Kim-strict in the sense of this paper. * In [ref] , Hanson studies a number of variants of Kim's Lemma which are related to ours.

    Bi-invariant types, reliably invariant types, and the comb tree property · Introduction · 0.99

  • While this paper was in preparation, two closely related preprints appeared. * In [ref] , Kim and Lee establish a different variant of Kim's Lemma for $$ theories.

    2211.04213 · Introduction · 0.83

  • A similar characterization of Kim-independence in the theory of a bilinear form on a vector space over an NSOP$_1$ field occurs in [ref] .

    2304.05404 · Bilinear forms over real closed fields · 0.93

Cites as background 1 paper
  • We say a formula $φ(x;b)$ universally Kim-divides[Universal Kim-dividing is called “strong Kim-dividing” in [ref] and “Conant-dividing” in [ref] .] over $M$ if it divides along Morley sequences for every global $M$-invariant type extending $(b/M)$.

    Conant-independence and generalized free amalgamation · Forking and dividing · 0.82

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