Hechler and Laver Trees

Arnold W. Miller

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arXiv · 1204.5198

Hechler and Laver Trees

Arnold W. Miller

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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

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BibTeX RIS

Arnold W. Miller. 2012-04-23. Hechler and Laver Trees. https://arxiv.org/abs/1204.5198

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