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1
Content available remote

On nonstructure of elementary submodels of a stable homogeneous structure

100%
Hyttinen T.
Fundamenta Mathematicae
|
1998
|
tom 156
|
nr 2
167-182
EN
We assume that M is a stable homogeneous model of large cardinality. We prove a nonstructure theorem for (slightly saturated) elementary submodels of M, assuming M has dop. We do not assume that th(M) is stable.
2
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Club-guessing and non-structure of trees

100%
Hyttinen T.
Fundamenta Mathematicae
|
2001
|
tom 168
|
nr 3
237-249
EN
We study the possibilities of constructing, in ZFC without any additional assumptions, strongly equivalent non-isomorphic trees of regular power. For example, we show that there are non-isomorphic trees of power ω₂ and of height ω · ω such that for all α < ω₁· ω · ω, E has a winning strategy in the Ehrenfeucht-Fraïssé game of length α. The main tool is the notion of a club-guessing sequence.
3
Content available remote

Metric abstract elementary classes with perturbations

64%
Hirvonen Å., Hyttinen T.
Fundamenta Mathematicae
|
2012
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tom 217
|
nr 2
123-170
EN
We define an abstract setting suitable for investigating perturbations of metric structures generalizing the notion of a metric abstract elementary class. We show how perturbation of Hilbert spaces with an automorphism and atomic Nakano spaces with bounded exponent fit into this framework, where the perturbations are built into the definition of the class being investigated. Further, assuming homogeneity and some other properties true in the example classes, we develop a notion of independence for this setting and show that it satisfies the usual independence axioms. Finally we define an isolation notion. Although it remains open whether this isolation gives any reasonable form of primeness, we prove that dominance works.
4
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Superstability in simple finitary AECs

64%
Hyttinen T., Kesälä M.
Fundamenta Mathematicae
|
2007
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tom 195
|
nr 3
221-268
EN
We continue the study of finitary abstract elementary classes beyond ℵ₀-stability. We suggest a possible notion of superstability for simple finitary AECs, and derive from this notion several good properties for independence. We also study constructible models and the behaviour of Galois types and weak Lascar strong types in this context. We show that superstability is implied by a-categoricity in a suitable cardinal. As an application we prove the following theorem: Assume that $(𝕂, ≼_𝕂)$ is a simple, tame, finitary AEC, a-categorical in some cardinal κ above the Hanf number such that cf(κ) > ω. Then $(𝕂, ≼_𝕂)$ is a-categorical in each cardinal above the Hanf number.
5
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Potential isomorphism and semi-proper trees

51%
Hellsten A., Hyttinen T., Shelah S.
Fundamenta Mathematicae
|
2002
|
tom 175
|
nr 2
127-142
EN
We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We show that the existence of weakly semi-proper trees is consistent relative to ZFC by proving the existence of weakly semi-proper trees under certain cardinal arithmetic assumptions. We also prove the consistency of the non-existence of weakly semi-proper trees assuming the consistency of some large cardinals.
6
Content available remote

More on the Ehrenfeucht-Fraisse game of length ω₁

51%
Hyttinen T., Shelah S., Vaananen J.
Fundamenta Mathematicae
|
2002
|
tom 175
|
nr 1
79-96
EN
By results of [9] there are models 𝔄 and 𝔅 for which the Ehrenfeucht-Fraïssé game of length ω₁, $EFG_{ω₁}(𝔄,𝔅)$, is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement "CH and $EFG_{ω₁}(𝔄,𝔅)$ is determined for all models 𝔄 and 𝔅 of cardinality ℵ₂" is that of a weakly compact cardinal. On the other hand, we show that if $2^{ℵ₀} < 2^{ℵ₃}$, T is a countable complete first order theory, and one of (i) T is unstable, (ii) T is superstable with DOP or OTOP, (iii) T is stable and unsuperstable and $2^{ℵ₀} ≤ ℵ₃$, holds, then there are 𝓐,ℬ ⊨ T of power ℵ₃ such that $EFG_{ω₁}(𝓐,ℬ)$ is non-determined.
7
Content available remote

On Borel reducibility in generalized Baire space

51%
Friedman S.-D., Hyttinen T., Kulikov V.
Fundamenta Mathematicae
|
2015
|
tom 231
|
nr 3
285-298
EN
We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $κ^{κ}$ for an uncountable κ with $κ^{<κ} = κ$. The theory looks quite different from its classical counterpart where κ = ω, although some basic theorems do generalize.
8
Content available remote

Model theory for infinite quantifier languages

38%
Hyttinen T.
Fundamenta Mathematicae
|
1990
|
tom 134
|
nr 2
125-142
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