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A classification of ordinals up to Borel isomorphism

Su Gao, Steve Jackson, Vincent Kieftenbeld (2008)

Fundamenta Mathematicae

We consider the Borel structures on ordinals generated by their order topologies and provide a complete classification of all ordinals up to Borel isomorphism in ZFC. We also consider the same classification problem in the context of AD and give a partial answer for ordinals ≤ω₂.

Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

Ramez Sami (1999)

Fundamenta Mathematicae

We prove the following theorem: Given a⊆ω and 1 α < ω 1 C K , if for some η < 1 and all u ∈ WO of length η, a is Σ α 0 ( u ) , then a is Σ α 0 . We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ 1 1 -Turing-determinacy implies the existence of 0 .

Bad Wadge-like reducibilities on the Baire space

Luca Motto Ros (2014)

Fundamenta Mathematicae

We consider various collections of functions from the Baire space ω ω into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on ω ω (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of...

Borel-Wadge degrees

Alessandro Andretta, Donald A. Martin (2003)

Fundamenta Mathematicae

Two sets of reals are Borel equivalent if one is the Borel pre-image of the other, and a Borel-Wadge degree is a collection of pairwise Borel equivalent subsets of ℝ. In this note we investigate the structure of Borel-Wadge degrees under the assumption of the Axiom of Determinacy.

Compact covering mappings and cofinal families of compact subsets of a Borel set

G. Debs, J. Saint Raymond (2001)

Fundamenta Mathematicae

Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement " α ω ω , ω L ( α ) < ω "; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets of a Borel...

Definable hereditary families in the projective hierarchy

R. Barua, V. Srivatsa (1992)

Fundamenta Mathematicae

We show that if ℱ is a hereditary family of subsets of ω ω satisfying certain definable conditions, then the Δ 1 1 reals are precisely the reals α such that β : α Δ 1 1 ( β ) . This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the Q 2 n + 1 -encodable reals is also shown.

Games with creatures

Saharon Shelah, Jindřich Zapletal (2003)

Commentationes Mathematicae Universitatis Carolinae

Many forcing notions obtained using the creature technology are naturally connected with certain integer games.

On partitions of lines and space

Paul Erdös, Steve Jackson, R. Mauldin (1994)

Fundamenta Mathematicae

We consider a set, L, of lines in n and a partition of L into some number of sets: L = L 1 . . . L p . We seek a corresponding partition n = S 1 . . . S p such that each line l in L i meets the set S i in a set whose cardinality has some fixed bound, ω τ . We determine equivalences between the bounds on the size of the continuum, 2 ω ω θ , and some relationships between p, ω τ and ω θ .

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