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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.

Borel extensions of Baire measures in ZFC

Menachem Kojman, Henryk Michalewski (2011)

Fundamenta Mathematicae

We prove: 1) Every Baire measure on the Kojman-Shelah Dowker space admits a Borel extension. 2) If the continuum is not real-valued-measurable then every Baire measure on M. E. Rudin's Dowker space admits a Borel extension. Consequently, Balogh's space remains the only candidate to be a ZFC counterexample to the measure extension problem of the three presently known ZFC Dowker spaces.

Cardinal and ordinal arithmetics of n -ary relational systems and n -ary ordered sets

Jiří Karásek (1998)

Mathematica Bohemica

The aim of this paper is to define and study cardinal (direct) and ordinal operations of addition, multiplication, and exponentiation for n -ary relational systems. n -ary ordered sets are defined as special n -ary relational systems by means of properties that seem to suitably generalize reflexivity, antisymmetry, and transitivity from the case of n = 2 or 3. The class of n -ary ordered sets is then closed under the cardinal and ordinal operations.

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