id: 03959408
dt: j
an: 03959408
au: Griffor, E.R.; Normann, D.
ti: Effective cofinalities and admissibility in E-recursion.
so: Fundam. Math. 123, 151-161 (1984).
py: 1984
pu: Instytut Matematyczny PAN, Warszawa
la: EN
cc: 
ut: E-closure; cofinality; admissibility; selection; extended plus one
    hypothesis; effective covering property; admissible sets
ci: 
li: 
ab: Given an ordinal $α$, let $E(α)$ be its E-closure. Kirousis proved that
    if $E(α)\vDash cf(\bar{\bar α})=ω$, then $E(α)$ is admissible.
    Motivated by this result, Sacks asked for a cofinality condition on
    $α$ that would characterize the admissibility of $E(α)$. The authors
    address this question, and show that while $Σ\sb 1$-admissibility of
    E($α)$ does not guarantee $cf(\bar{\bar α})=ω$ in $E(α)$, it does
    imply the weaker condition that the RE$\wedge co$-RE cofinality of $α$
    (which one may assume to be the greatest cardinal of $E(α)$) is $ω$.
    By definition the $RE\wedge co$-RE cofinality of $α$ is the least
    $τ\le α$ for which there exists a set $R\subseteq α$ of order type
    $τ$ such that R is the intersection of an RE set on a co-RE set. The
    second main result of this paper is a dynamic proof of selection, so
    that on $E(α)$ there is a uniform selection over RE subsets of each
    $δ<cf\sp{E(α)}(α)$. This gives as a corollary the consistency of the
    extended plus one hypothesis at the type 3 level with not CH. Some
    applications of the selection result are presented, including an
    effective covering property that any co-RE subset of $cf\sp{E(α)}(α)$
    can be covered by a REC set of the same order type. This paper
    concludes with an elementary proof, through a study of methods from
    Girard’s $β$-logic, of Van de Wiele’s result that every uniformly
    $Σ\sb 1$-definable function on the universe of sets, and total over
    all admissible sets, is E-recursive.
rv: C.T.Chong