1,667 research outputs found

    On Martin's Pointed Tree Theorem

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    We investigate the reverse mathematics strength of Martin's pointed tree theorem (MPT) and one of its variants, weak Martin's pointed tree theorem (wMPT)

    Preserving levels of projective determinacy by tree forcings

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    We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.Comment: 3 figure

    Laver and set theory

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    In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

    Lebesgue's Density Theorem and definable selectors for ideals

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    We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio
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