A Course in Model Theory

Poizat, Bruno

doi:10.1007/978-1-4419-8622-1

Cited by 163 publications

(75 citation statements)

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“…Our proof of Theorem 2.1 uses the following fact due to a combination of Shelah and Poizat. Details are given in chapter 12 of [7]. (1) for every M ≺ N |= T such that both models are sufficiently saturated, there are no more than 2 |M | complete one-types q(x) over N such that q(x) is finitely realisable in M , (2) for every finite n ≥ 1 and every M ≺ N |= T such that both models are sufficiently saturated, there are no more than 2 |M | complete n-types q(x) over N such that q(x) is finitely realisable in M .…”

Section: A General Resultsmentioning

confidence: 99%

NIP for some pair-like theories

Boxall

2010

Arch. Math. Logic

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Section: A General Resultsmentioning

confidence: 99%

NIP for some pair-like theories

Boxall

2010

Arch. Math. Logic

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“…A well-known equivalence (see Theorem 12.17 of [25]) is We now give some consequences of the NIP for Keisler measures. The main insight is due to Keisler ([13], Theorem 3.14).…”

Section: Nip and Some Consequencesmentioning

confidence: 97%

Groups, measures, and the NIP

Hrushovski

Peterzil

Pillay

2007

J. Amer. Math. Soc.

185

354

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We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups G in saturated o-minimal structures to compact Lie groups. We also prove some other structural results about such G, for example the existence of a left invariant finitely additive probability measure on definable subsets of G. We finally introduce a new notion "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the o-minimal case.

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“…(Thorn forking is of course a very useful notion but does not apply to the C-minimal case.) This section builds on work of Poizat [34], Shelah [36] and Adler [1]. Many of our key notions make an explicit or implicit appearance in Chapter 12 of the Poizat reference.…”

Section: Forking and Lascar Strong Typesmentioning

confidence: 99%

On NIP and invariant measures

Hrushovski¹,

Pillay²

2011

J. Eur. Math. Soc.

151

269

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Abstract. We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of "generically stable" types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o-minimal expansions of real closed fields.

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A Course in Model Theory

Cited by 163 publications

References 0 publications

NIP for some pair-like theories

NIP for some pair-like theories

Groups, measures, and the NIP

On NIP and invariant measures

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