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### Countable partition ordinals

Rene Schipperus
2010 Annals of Pure and Applied Logic
The structure of ordinals of the form ω ω β for countable β is studied. The main result is: Theorem 1.  ...  If β < ω 1 is the sum of one or two indecomposable ordinals, then Also an example is given to show that α → (α, 3) 2 need not imply α → (α, n) 2 for all n < ω.  ...  is widespread for countable ordinals.  ...

Juichi Shinoda
1985 Nagoya mathematical journal
In , it also has been shown that an ordinal &gt; ω is-admissible if it is |a|S-recursively inaccessible, where |a|s- is the ordinal denoted bya.  ...  In , Platek constructs a hierarchy of jumpsindexed by elements a of a setof ordinal notations. He asserts that a realX⊆ ω is recursive in the superjumpSif and only if it is recursive in some.  ...  In  , it has been shown that \(P S \ = 4 Let a > ω be a countable admissible ordinal.  ...

### The Church Problem for Countable Ordinals

Alexander Rabinovich, Wolfgang Thomas
2009 Logical Methods in Computer Science
We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length.  ...  We prove that determinacy and decidability parts of the Bu}chi and Landweber theorem hold for all countable ordinals and that its full extension holds for all ordinals < \omega\^\omega.  ...  Büchi generalized the concept of an automaton to allow automata to "work" on words of any countable length (ordinal) and used this to show that the MLO-theory of any countable ordinal is decidable (see  ...

### Computability on the space of countable ordinals [article]

Arno Pauly
2017 arXiv   pre-print
While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far.  ...  We propose a notion of computability on the space of countable ordinals via a representation in the sense of computable analysis.  ...  successor ordinal, and every countable successor ordinal occurs as the height of a nice relation).  ...

### Countable Admissible Ordinals and Hyperdegrees [chapter]

GERALD E. SACKS
1999 World Scientific Series in 20th Century Mathematics
Let 01 be the least ordinal not countable in A. Clearly 01 is admissible.  ...  By 4.7 L[oL, K] is a countable admissible set that satisfies & dependent choice, and (y. is the least ordinal not countable in L[ol, K].  ...

### Countable Box Products of Ordinals

Mary Ellen Rudin
1974 Transactions of the American Mathematical Society
The countable box product of ordinals is examined in the paper for normality and paracompactness.  ...  The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal.  ...  The continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact.  ...

### Countable box products of ordinals

Mary Ellen Rudin
1974 Transactions of the American Mathematical Society
The countable box product of ordinals is examined in the paper for normality and paracompactness.  ...  The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal.  ...  The continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact.  ...

### Countable admissible ordinals and hyperdegrees

Gerald E Sacks
Let 01 be the least ordinal not countable in A. Clearly 01 is admissible.  ...  By 4.7 L[oL, K] is a countable admissible set that satisfies & dependent choice, and (y. is the least ordinal not countable in L[ol, K].  ...

### Pairs of orthogonal countable ordinals [article]

Claude Laflamme, Maurice Pouzet, Nobert Sauer, Imed Zaguia
2014 arXiv   pre-print
We characterize pairs of orthogonal countable ordinals.  ...  We prove that if α and β are two countable ordinals, with α≤β, then α and β are orthogonal if and only if either ω + 1≤α or α =ω and β < ωβ.  ...  An ordinal α is orthogonal to ω if and only if α is countably infinite and ωα > α.  ...

### Countable ordinal spaces and compact countable subsets of a metric space [article]

Borys Álvarez-Samaniego, Andrés Merino
2018 arXiv   pre-print
We show in detail that every compact countable subset of a metric space is homeomorphic to a countable ordinal number, which extends a result given by Mazurkiewicz and Sierpinski for finite-dimensional  ...  In addition, we prove that for all metric space (E,d), the cardinality of the set of all the equivalence classes K_E, up to homeomorphisms, of compact countable subsets of E is less than or equal to _1  ...  Let (E, d) be a metric space and let α > 1 be a countable ordinal number.  ...

### Forcing with copies of countable ordinals

Miloš S. Kurilić
2014 Proceedings of the American Mathematical Society
Let α be a countable ordinal and P(α) the collection of its subsets isomorphic to α.  ...  Also we analyze the quotients over ordinal ideals P (ω δ )/I ω δ and the corresponding cardinal invariants h ω δ and t ω δ .  ...  Let δ, δ > 0 be countable ordinals. Then (a) The ordinal ω δ is an ω-sum of elements of H satisfying (2.1); MILO Š S. KURILI Ć that A ⊂ * I B and A \ B ∈ I G .  ...

### Forcing With Copies of Countable Ordinals

Milos Kurilic
2013 arXiv   pre-print
Let \alpha be a countable ordinal and \P(\alpha) the collection of its subsets isomorphic to \alpha.  ...  or 1 and I(\omega ^\gamma) the corresponding ordinal ideal.  ...  Fact 2.4 For a countable limit ordinal α the following conditions are equivalent: (a) α is indecomposable (i.e. α is not a sum of two smaller ordinals); (b) β + γ < α, for each β, γ < α; (c) A ∈ P(α) or  ...

### A Subset of the Countable Ordinals

Mary Ellen Rudin
1957 The American mathematical monthly
A SUBSET OF THE COUNTABLE ORDINALS Mary ELten Rubin, University of Rochester Let A be the space of all countable ordinals with the order topology.  ...  For each positive integer n, let X, be a countable collection of intervals covering L of length less than 1/m such that, for »>1, X, is a refinement of X,_1.  ...

### First-order separation over countable ordinals [article]

Thomas Colcombet, Sam van Gool, Rémi Morvan
2022 arXiv   pre-print
We show that the existence of a first-order formula separating two monadic second order formulas over countable ordinal words is decidable.  ...  We also show the computability of FO-pointlike sets, and the decidability of the covering problem for first-order logic on countable ordinal words.  ...  Countable ordinal words are sequences of letters that are indexed by a countable total wellordering, i.e., up to isomorphism, by a countable ordinal.  ...

### Countable ordinals and big Ramsey degrees [article]

Dragan Mašulović, Branislav Šobot
2019 arXiv   pre-print
In this paper we consider big Ramsey degrees of finite chains in countable ordinals. We prove that a countable ordinal has finite big Ramsey degrees if and only if it is smaller than ω^ω.  ...  Big Ramsey degrees of finite chains in all other countable ordinals are infinite.  ...  Let α be a countable ordinal with finite big Ramsey degrees.  ...
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