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Trees and -subsets of ω1ω1

Published online by Cambridge University Press:  12 March 2014

Alan Mekler  and
Jouko Väänänen
Alan Mekler
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, Canada
Jouko Väänänen
Affiliation:
Department of Mathematics, University of Helsinki, Helsinki, Finland
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Abstract

We study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of .

We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space .

We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2).

Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 3 , September 1993 , pp. 1052 - 1070
Copyright
Copyright © Association for Symbolic Logic 1993

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