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Countable partition ordinals

2010
*
Annals of Pure and Applied Logic
*

The structure of

doi:10.1016/j.apal.2009.12.007
fatcat:rvrxvanie5cgdd4lequ25effkm
*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*. ...##
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Countable -admissible Ordinals

1985
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Nagoya mathematical journal
*

In [1], it also has been shown that an

doi:10.1017/s0027763000021450
fatcat:smwti4m2xjc33kr55immfmxztm
*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*. ...##
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The Church Problem for Countable Ordinals

2009
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Logical Methods in Computer Science
*

We consider a natural generalization of the Church problem to

doi:10.2168/lmcs-5(2:5)2009
fatcat:nlikkkg23zgfhhvavn2a2bur6q
*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 ...##
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Computability on the space of countable ordinals
[article]

2017
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arXiv
*
pre-print

While there is a well-established notion of what a computable

arXiv:1501.00386v3
fatcat:3tsimzsrlrhzjhhgabxkkywbui
*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). ...##
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Countable Admissible Ordinals and Hyperdegrees
[chapter]

1999
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World Scientific Series in 20th Century Mathematics
*

Let 01 be the least

doi:10.1142/9789812812926_0015
fatcat:inicmpvwbjcdredbencukybslm
*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]. ...##
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Countable Box Products of Ordinals

1974
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Transactions of the American Mathematical Society
*

The

doi:10.2307/1996824
fatcat:sp37z5hfgvgjnptyagt2owdk6a
*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. ...##
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Countable box products of ordinals

1974
*
Transactions of the American Mathematical Society
*

The

doi:10.1090/s0002-9947-1974-0340022-1
fatcat:tccfzh62svdhrprinjxkv34kqi
*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. ...##
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Countable admissible ordinals and hyperdegrees

1976
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Advances in Mathematics
*

Let 01 be the least

doi:10.1016/0001-8708(76)90187-0
fatcat:4qere2zfqfb4fpcwgktsas66b4
*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]. ...##
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Pairs of orthogonal countable ordinals
[article]

2014
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arXiv
*
pre-print

We characterize pairs of orthogonal

arXiv:1407.0940v1
fatcat:sylth3lkqbc77fajmhb6atb6a4
*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. ...##
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Countable ordinal spaces and compact countable subsets of a metric space
[article]

2018
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arXiv
*
pre-print

We show in detail that every compact

arXiv:1803.00400v2
fatcat:agxazxb6efaqxdvfefv2pwaxom
*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. ...##
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Forcing with copies of countable ordinals

2014
*
Proceedings of the American Mathematical Society
*

Let α be a

doi:10.1090/s0002-9939-2014-12360-4
fatcat:htea34qglrcyzhq5hrsoew37am
*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 . ...##
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Forcing With Copies of Countable Ordinals

2013
*
arXiv
*
pre-print

Let \alpha be a

arXiv:1304.7714v1
fatcat:eb2rt6lplzfc5fcxp664xu625i
*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 ...##
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A Subset of the Countable Ordinals

1957
*
The American mathematical monthly
*

A SUBSET OF THE

doi:10.2307/2309601
fatcat:puvtw7gnwjbyzepnn63zv72bqy
*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. ...##
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First-order separation over countable ordinals
[article]

2022
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arXiv
*
pre-print

We show that the existence of a first-order formula separating two monadic second order formulas over

arXiv:2201.03089v1
fatcat:k4nolvsumjc4rmjdw2o2wrev24
*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*. ...##
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Countable ordinals and big Ramsey degrees
[article]

2019
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arXiv
*
pre-print

In this paper we consider big Ramsey degrees of finite chains in

arXiv:1904.03637v2
fatcat:unkvtjp6lveoponhwdu4jdmqoq
*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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