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.
Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M.Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes.Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.
A canary tree is a tree of cardinality the continuum which has no uncountable branch, but gains a branch whenever a stationary set is destroyed (without adding reals). Canary trees are important in infinitary model theory. The existence of a canary tree is independent of ZFC + GCH.
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