L'égalité au cube

Poizat, Bruno

doi:10.2307/2694967

Cited by 38 publications

(73 citation statements)

References 7 publications

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“…It may also be worth noting that if for every finite A there exists a finite B such that A B then the above assumption will always hold. For a discussion of a construction where this is not the case, see [Poi01] (where in fact the self-sufficient closure is not well defined).…”

Section: From Envelopes To Pseudo-envelopesmentioning

confidence: 99%

Some questions concerning Hrushovski's amalgamation constructions

Hasson

2008

J. Inst. Math. Jussieu

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In order to construct a counterexample to Zilber's conjecture—that a strongly minimal set has a degenerate, affine or field-like geometry—Ehud Hrushovski invented an amalgamation technique which has yielded all the exotic geometries so far. We shall present a framework for this construction in the language of standard geometric stability and show how some of the recent constructions fit into this setting. We also ask some fundamental questions concerning this method.

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Section: From Envelopes To Pseudo-envelopesmentioning

confidence: 99%

Some questions concerning Hrushovski's amalgamation constructions

Hasson

2008

J. Inst. Math. Jussieu

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“…In the first case, the subgroup is in the divisible hull of a finitely generated group (-structures of Lang type). In the second case, we consider structures constructed via a Hrushovski predimension as in Poizat's 'field with green points' from [9] (-structures of Poizat type). The first case is relevant to Section 2 in Zilber's paper [13] and the second case is relevant to Sections 3 and 4 there.…”

Section: (I) (A(0) = 0) ∧ (∀T)((t = 0) → γ(A(t))); (Ii) (∀T)(∀g)(g(g)mentioning

confidence: 99%

“…We wish to consider Poizat's 'field with green points' (F, G(F )) from [9]. This is a structure in a language L for fields with an extra unary predicate G. The field F is algebraically closed of characteristic zero and the subset G(F ) (-the 'green points') is a torsion free divisible subgroup of the multiplicative group F × .…”

Section: 2mentioning

confidence: 99%

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Expansions of fields by angular functions

Evans¹

2008

J. Inst. Math. Jussieu

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Abstract. The notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two 'counterexamples' from model theory: the nonclassical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function.

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“…We say K has δ-formulas for minimal intrinsic extensions if for each pair (B, A) with B minimal intrinsic over A, there is a formula φ AB (x, y) such that if φ AB (b , a ) and B , A are the structures generated b , a then δ(B /A ) ≤ δ(B/A) (and some other conditions we won't spell out here). The existence of δ formulas is trivial in the ab initio case [25], routine for bicolored fields [7] and impossible (in full generality) for fields with a distinguished multiplicative subgroup [22,40].…”

Section: We Say B Is a Minimal Intrinsic Extension Of A If δ(B/a) < 0mentioning

confidence: 99%

Notes on Quasiminimality and Excellence

Baldwin¹

2004

Bull. symb. log.

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Zilber's proposes [60] to prove 'canonicity results for pseudo-analytic' structures. Informally, 'canonical means the theory of the structure in a suitable possibly infinitary language (see Section 2) has one model in each uncountable power' while 'pseudoanalytic means the model of power 2 ℵ0 can be taken as a reduct of an expansion of the complex numbers by analytic functions'. This program interacts with two other lines of research. First is the general study of categoricity theorems in infinitary languages. After initial results by Keisler, reported in [31], this line was taken up in a long series of works by Shelah. We place Zilber's work in this context. The second direction stems from Hrushovski's construction of a counterexample to Zilber's conjecture that every strongly minimal set is 'trivial', 'vector space-like', or 'field-like'. This construction turns out to be very concrete example of the Abstract Elementary Classes which arose in Shelah's analysis. This paper examines the intertwining of these three themes.The study of (C, +, ·, exp) leads one immediately to some extension of first order logic; the integers with all their arithmetic are first order definable in (C, +, ·, exp). Thus, the first order theory of complex exponentiation is horribly complicated; it is certainly unstable and so can't be first order categorical. One solution is to use infinitary logic to pin down the pathology. Insist that the kernel of the exponential map is fixed as a single copy of the integers while allowing the rest of the structure to grow. We describe in Section 5 Zilber's program to show, modulo certain (very serious) algebraic hypotheses, that (C, +, ·, exp) can be axiomatized by a categorical L ω 1 ,ω -sentence.Of course, the extension from first order logic causes the failure of the compactness theorem. (E.g., it is easy to write a sentence in L ω 1 ,ω whose only model is the natural numbers with successor). But there are some more subtle losses. In first order logic, a type can be given as a syntactic object -a consistent set of formulas. Consider the theory T of a dense linear order without endpoints, a unary predicate P (x) which is dense and codense, and an infinite set of constants arranged in order type ω + ω * . Let K be class of all models of T which omit the type of a pair of points, which are both in the cut determined by the constants. Now consider the types p and q which are satisfied by a point in the cut, which is in P or in ¬P respectively. Now p and q are each satisfiable in a member of K but they are not simultaneously satisfiable. This failure of amalgamation shows that a more subtle notion than consistency is needed to describe types.We took 'canonical' above as meaning 'categorical in uncountable cardinalities'. The analysis of first order theories categorical in power is based on first studying strongly minimal sets: every definable subset is finite or cofinite. A natural generalization of this, particularly since it holds of simply defined subsets of (C, +, ·, exp), is to consider structur...

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L'égalité au cube

Cited by 38 publications

References 7 publications

Some questions concerning Hrushovski's amalgamation constructions

Some questions concerning Hrushovski's amalgamation constructions

Expansions of fields by angular functions

Notes on Quasiminimality and Excellence

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