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  • 1
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    Berlin : Mayo Foundation for Medical Education and Research
    Keywords: Medicine ; Dermatology ; Family medicine ; Geriatrics ; Internal Medicine ; Oncology ; Medicine & Public Health ; Dermatology ; General Practice / Family Medicine ; Oncology ; Internal Medicine ; Geriatrics/Gerontology ; Springer eBooks
    Pages: : digital
    ISBN: 9783642133992
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  • 2
    ISSN: 1432-0665
    Keywords: 03E35 ; 06E99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results: (1) Ifθ = 〈κ α :α 〈ω 1〉 is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB. (2) Ifκ is an uncountable cardinal such thatκ 〈κ =κ andθ = 〈κ α :α 〈κ +〉 is a cardinal sequence such thatκ α ≥κ for everyα 〈κ + andκ α =κ for everyα 〈κ + such that cf(α)〈κ, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebraB such that θ is the cardinal sequence ofB.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-0665
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. The countable sequences of cardinals which arise as cardinal sequences of superatomic Boolean algebras were characterized by La Grange on the basis of ZFC set theory. However, no similar characterization is available for uncountable cardinal sequences. In this paper we prove the following two consistency results: (1) If $\theta =\langle\kappa_{\alpha}:\alpha 〈 \omega_1 \rangle$ is a sequence of infinite cardinals, then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebra $B$ such that $\theta$ is the cardinal sequence of $B$ . (2) If $\kappa$ is an uncountable cardinal such that $\kappa^{〈 \kappa} = \kappa$ and $\theta = \langle\kappa_{\alpha}:\alpha 〈 \kappa^+ \rangle$ is a cardinal sequence such that $\kappa_{\alpha} \geq\kappa$ for every $\alpha 〈 \kappa^+$ and $\kappa_{\alpha} =\kappa$ for every $\alpha 〈 \kappa^+$ such that $\mbox{cf}(\alpha) 〈 \kappa$ , then there is a cardinal-preserving notion of forcing that changes cardinal exponentiation and forces the existence of a superatomic Boolean algebra $B$ such that $\theta$ is the cardinal sequence of $B$ .
    Type of Medium: Electronic Resource
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