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On Borel reducibility in generalized Baire space

Sy-David Friedman, Tapani Hyttinen, Vadim Kulikov (2015)

Fundamenta Mathematicae

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.

On convergence of integrals in o-minimal structures on archimedean real closed fields

Tobias Kaiser (2005)

Annales Polonici Mathematici

We define a notion of volume for sets definable in an o-minimal structure on an archimedean real closed field. We show that given a parametric family of continuous functions on the positive cone of an archimedean real closed field definable in an o-minimal structure, the set of parameters where the integral of the function converges is definable in the same structure.

On definably proper maps

Mário J. Edmundo, Marcello Mamino, Luca Prelli (2016)

Fundamenta Mathematicae

In this paper we work in o-minimal structures with definable Skolem functions, and show that: (i) a Hausdorff definably compact definable space is definably normal; (ii) a continuous definable map between Hausdorff locally definably compact definable spaces is definably proper if and only if it is a proper morphism in the category of definable spaces. We give several other characterizations of definably proper, including one involving the existence of limits of definable types. We also prove the...

On d-finite tuples in random variable structures

Shichang Song (2013)

Fundamenta Mathematicae

We prove that the d-finite tuples in models of ARV are precisely the discrete random variables. Then, we apply d-finite tuples to the work by Keisler, Hoover, Fajardo, and Sun concerning saturated probability spaces. In particular, we strengthen a result in Keisler and Sun's recent paper.

On d-finiteness in continuous structures

Itaï Ben Yaacov, Alexander Usvyatsov (2007)

Fundamenta Mathematicae

We observe that certain classical results of first order model theory fail in the context of continuous first order logic. We argue that this happens since finite tuples in a continuous structure may behave as infinite tuples in classical model theory. The notion of a d-finite tuple attempts to capture some aspects of the classical finite tuple behaviour. We show that many classical results involving finite tuples are valid in continuous logic upon replacing "finite" with "d-finite". Other results,...

On divisibility in definable groups

Margarita Otero (2009)

Fundamenta Mathematicae

Let ℳ be an o-minimal expansion of a real closed field. It is known that a definably connected abelian group is divisible. We show that a definably compact definably connected group is divisible.

On embedding models of arithmetic of cardinality ℵ₁ into reduced powers

Juliette Kennedy, Saharon Shelah (2003)

Fundamenta Mathematicae

In the early 1970’s S. Tennenbaum proved that all countable models of PA₁¯ + ∀₁ -Th(ℕ) are embeddable into the reduced product ℕ ω / ℱ , where ℱ is the cofinite filter. In this paper we show that if M is a model of PA¯ + ∀₁ - Th(ℕ), and |M| = ℵ₁, then M is embeddable into ℕ ω / D , where D is any regular filter on ω.

On equivalence relations second order definable over H(κ)

Saharon Shelah, Pauli Vaisanen (2002)

Fundamenta Mathematicae

Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 second order definable over H(κ) if there exists a second order sentence ϕ and a parameter P ⊆ H(κ) such that functions f and g from κ into 2 are equivalent iff the structure ⟨ H(κ), ∈, P, f, g ⟩ satisfies ϕ. The possible numbers of equivalence classes of second order definable equivalence relations include all the nonzero cardinals at most κ⁺. Additionally, the possibilities are closed under unions...

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