The model N = ∪ {L[A]: A countable set of ordinals}
نویسندگان
چکیده
منابع مشابه
Countable 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.
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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 کاملPartitioning Pairs of Countable Sets of Ordinals
Lemma 1. There are ω1 disjoint stationary subsets of ω1. Proof. Let Cub(ω1) = {X ⊆ ω1 : ∃C ⊆ X(C is club on ω1)}. Then Cub(ω1) is countably complete filter. Its dual ideal Cub (ω1) = {X ⊆ ω1 : ∃X ′ ∈ Cub(ω1)(X = ω1\X ′)} is countably complete and contains all singletons and so all countable subsets of ω1. Recall also that X ⊆ ω1 is stationary if and only if X / ∈ Cub(ω1). For every ρ < ω1 let f...
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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 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....
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ژورنال
عنوان ژورنال: Annals of Pure and Applied Logic
سال: 1987
ISSN: 0168-0072
DOI: 10.1016/0168-0072(87)90021-2