Countable ordinals and the analytical hierarchy. I
نویسندگان
چکیده
منابع مشابه
Countable Ordinals and the Analytical Hierarchy, I
Proof. For notational simplicity let us take n = 1 as a typical case. Thus let A C WO be ΣJ and assume sup {|a |: a E A} < Nj. Let B Qω be Π2 and /: ω ->ω recursive such that f[B] = A. Consider then the following game: Player I plays β, player II plays γ and II wins iff γ E WO&(β E B -> \f(β)| g | γ | ) . Clearly player II has a winning strategy in this game. But his payoff set is Σ\9 so by a r...
متن کاملCountable Ordinals
This development defines a well-ordered type of countable ordinals. It includes notions of continuous and normal functions, recursively defined functions over ordinals, least fixed-points, and derivatives. Much of ordinal arithmetic is formalized, including exponentials and logarithms. The development concludes with formalizations of Cantor Normal Form and Veblen hierarchies over normal functions.
متن کاملImages of the Countable Ordinals
We study spaces that are continuous images of the usual space [0, ω1) of countable ordinals. We begin by showing that if Y is such a space and is T3 then Y has a monotonically normal compactification, and is monotonically normal, locally compact and scattered. Examples show that regularity is needed in these results. We investigate when a regular continuous image of the countable ordinals must ...
متن کاملWalks on Countable Ordinals and Selective Ultrafilters
In our previous work we have introduced filters on the set of countable ordinals as invariants to standard characteristics of walks in this domain. In this note we examine their projections to the set of natural numbers.
متن کاملThe Church Problem for Countable Ordinals
A fundamental theorem of Büchi and Landweber shows that the Church synthesis problem is computable. Büchi and Landweber reduced the Church Problem to problems about ω-games and used the determinacy of such games as one of the main tools to show its computability. We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length....
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 1975
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.1975.60.223