Hechler and Laver Trees
Logic
2012-04-25 v1
Abstract
A Laver tree is a tree in which each node splits infinitely often. A Hechler tree is a tree in which each node splits cofinitely often. We show that every analytic set is either disjoint from the branches of a Heckler tree or contains the branches of a Laver tree. As a corollary we deduce Silver Theorem that all analytic sets are Ramsey. We show that in Godel's constructible universe that our result is false for co-analytic sets (equivalently it fails for analytic sets if we switch Hechler and Laver). We show that under Martin's axiom that our result holds for Sigma^1_2 sets. Finally we define two games related to this property. Latex2e 8 pages Latest version at http://www.math.wisc.edu/~miller/res/index.html
Keywords
Cite
@article{arxiv.1204.5198,
title = {Hechler and Laver Trees},
author = {Arnold W. Miller},
journal= {arXiv preprint arXiv:1204.5198},
year = {2012}
}