Andrés E. Caicedo

BPFA and Projective Well-Orderings of the Reals

Andrés Eduardo Caicedo et al. — Mon, 13 Feb 2012 14:55:29 PST

If the bounded proper forcing axiom BPFA holds and ω₁=ω₁^L, then there is a lightface Σ¹₃ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ¹₄, for many "consistently locally certified" relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ₂ stable ordinals of L.

Set Theory and Its Applications: Annual Boise Extravaganza in Set Theory, 1995--2010, Boise, Idaho

Liljana Babinkostova et al. — Fri, 15 Jul 2011 09:23:02 PDT

This book consists of several survey and research papers covering a wide range of topics in active areas of set theory and set theoretic topology. Some of the articles present, for the first time in print, knowledge that has been around for several years and known intimately to only a few experts. The surveys bring the reader up to date on the latest information in several areas that have been surveyed a decade or more ago. Topics covered in the volume include combinatorial and descriptive set theory, determinacy, iterated forcing, Ramsey theory, selection principles, set-theoretic topology, and universality, among others. Graduate students and researchers in logic, especially set theory, descriptive set theory, and set-theoretic topology, will find this book to be a very valuable reference.

Intersecting Families and Definability

Andrés E. Caicedo — Wed, 21 Apr 2010 13:28:50 PDT

Multiboard Determinacy

Andrés E. Caicedo — Fri, 16 Apr 2010 11:13:20 PDT

Regressive Functions on Pairs

Andrés Eduardo Caicedo — Wed, 07 Apr 2010 13:19:29 PDT

We compute an explicit upper bound for the regressive Ramsey numbers by a combinatorial argument, the corresponding function being of Ackermannian growth. For this, we look at the more general problem of bounding g(n,m), the least l such that any regressive function ƒ: [m; l]^[2]—>N admits a min-homogeneous set of size n. Analysis of this function also leads to the simplest known proof that the regressive Ramsey numbers have rate of growth at least Ackermannian. Together, these results give a purely combinatorial proof that, for each m, g(·,m) has rate of growth precisely Ackermannian, considerably improve the previously known bounds on the size of regressive Ramsey numbers, and provide the right rate of growth of the levels of g. For small numbers we also nd bounds on their value under g improving the ones provided by our general argument.