Trees and $Pi^11$-Subsets of $^{omega1}omega1$
Alan H. Mekler, Jouko Väänánen · Project Euclid (Cornell University) · 1993
We study descriptive set theory in the space $^{\\omega_1}\\omega_1$ 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 $\\Pi^1_1$-sets of $^{\\omega_1}\\omega_1$. We call a family $\\mathscr{U}$ of trees universal for a class $\\mathscr{V}$ of trees if $\\mathscr{U} \\subseteq \\mathscr{V}$ and every tree in $\\mathscr{V}$ can be order-preservingly mapped into a tree in $\\mathscr{U}$. It is well known that the class of countable trees with no infinite branches has a universal family of size $\\aleph_1$. We shall study the smallest cardinality of a universal family for the class of trees of cardinality $\\leq\\aleph_1$ with no uncountable branches. We prove that this cardinality can be 1 (under $\ eg$CH) and any regular cardinal $\\kappa$ which satisfies $\\aleph_2 \\leq \\kappa \\leq 2^{\\aleph_1}$ (under CH). This bears immediately on the covering property of the $\\Pi^1_1$-subsets of the space $^{\\omega_1}\\omega_1$. We also study the possible cardinalities of definable subsets of $^{\\omega_1}\\omega_1$. We show that the statement that every definable subset of $^{\\omega_1}\\omega_1$ has cardinality $<\\omega_n$ or cardinality $2^{\\omega_1}$ is equiconsistent with ZFC (if $n \\geq 3$) and with ZFC plus an inaccessible (if $n = 2$). Finally, we define an analogue of the notion of a Borel set for the space $^{\\omega_1}\\omega_1$ and prove a Souslin-Kleene type theorem for this notion.