1,667 research outputs found
On Martin's Pointed Tree Theorem
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
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
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
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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