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Countable partition ordinals
2010
Annals of Pure and Applied Logic
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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. ...
doi:10.1016/j.apal.2009.12.007
fatcat:rvrxvanie5cgdd4lequ25effkm
Countable -admissible Ordinals
1985
Nagoya mathematical journal
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In [1], it also has been shown that an ordinal > ω is-admissible if it is |a|S-recursively inaccessible, where |a|s- is the ordinal denoted bya. ...
In [3], 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 [1] , it has been shown that \(P S \ = 4 Let a > ω be a countable admissible ordinal. ...
doi:10.1017/s0027763000021450
fatcat:smwti4m2xjc33kr55immfmxztm
The Church Problem for Countable Ordinals
2009
Logical Methods in Computer Science
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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 ...
doi:10.2168/lmcs-5(2:5)2009
fatcat:nlikkkg23zgfhhvavn2a2bur6q
Computability on the space of countable ordinals
[article]
2017
arXiv
pre-print
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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). ...
arXiv:1501.00386v3
fatcat:3tsimzsrlrhzjhhgabxkkywbui
Countable Admissible Ordinals and Hyperdegrees
[chapter]
1999
World Scientific Series in 20th Century Mathematics
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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]. ...
doi:10.1142/9789812812926_0015
fatcat:inicmpvwbjcdredbencukybslm
Countable Box Products of Ordinals
1974
Transactions of the American Mathematical Society
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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. ...
doi:10.2307/1996824
fatcat:sp37z5hfgvgjnptyagt2owdk6a
Countable box products of ordinals
1974
Transactions of the American Mathematical Society
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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. ...
doi:10.1090/s0002-9947-1974-0340022-1
fatcat:tccfzh62svdhrprinjxkv34kqi
Countable admissible ordinals and hyperdegrees
1976
Advances in Mathematics
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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]. ...
doi:10.1016/0001-8708(76)90187-0
fatcat:4qere2zfqfb4fpcwgktsas66b4
Pairs of orthogonal countable ordinals
[article]
2014
arXiv
pre-print
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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. ...
arXiv:1407.0940v1
fatcat:sylth3lkqbc77fajmhb6atb6a4
Countable ordinal spaces and compact countable subsets of a metric space
[article]
2018
arXiv
pre-print
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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. ...
arXiv:1803.00400v2
fatcat:agxazxb6efaqxdvfefv2pwaxom
Forcing with copies of countable ordinals
2014
Proceedings of the American Mathematical Society
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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 . ...
doi:10.1090/s0002-9939-2014-12360-4
fatcat:htea34qglrcyzhq5hrsoew37am
Forcing With Copies of Countable Ordinals
2013
arXiv
pre-print
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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 ...
arXiv:1304.7714v1
fatcat:eb2rt6lplzfc5fcxp664xu625i
A Subset of the Countable Ordinals
1957
The American mathematical monthly
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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. ...
doi:10.2307/2309601
fatcat:puvtw7gnwjbyzepnn63zv72bqy
First-order separation over countable ordinals
[article]
2022
arXiv
pre-print
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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. ...
arXiv:2201.03089v1
fatcat:k4nolvsumjc4rmjdw2o2wrev24
Countable ordinals and big Ramsey degrees
[article]
2019
arXiv
pre-print
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2019 pre-print arXiv:1904.03637v1 fatcat:old6eod6ejcfnos4te3qljwknm
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. ...
arXiv:1904.03637v2
fatcat:unkvtjp6lveoponhwdu4jdmqoq
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