Bohr compactifications of discrete structures

Joan Hart; Kenneth Kunen

Fundamenta Mathematicae (1999)

  • Volume: 160, Issue: 2, page 101-151
  • ISSN: 0016-2736

Abstract

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We prove the following theorem: Given a⊆ω and 1 ≤ α < ω 1 C K , if for some η < ℵ 1 and all u ∈ WO of length η, a is Σ α 0 ( u ) , then a is Σ α 0 .We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: Σ 1 1 -Turing-determinacy implies the existence of 0 .

How to cite

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Hart, Joan, and Kunen, Kenneth. "Bohr compactifications of discrete structures." Fundamenta Mathematicae 160.2 (1999): 101-151. <http://eudml.org/doc/212384>.

@article{Hart1999,
abstract = {We prove the following theorem: Given a⊆ω and $1 ≤ α < ω_1^\{CK\}$, if for some $η < ℵ_1$ and all u ∈ WO of length η, a is $Σ _α^0(u)$, then a is $Σ_α^0$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: $Σ_1^1 $-Turing-determinacy implies the existence of $0^\{#\}$. },
author = {Hart, Joan, Kunen, Kenneth},
journal = {Fundamenta Mathematicae},
keywords = {-structure; quasigroup; loop; Bohr compactifications},
language = {eng},
number = {2},
pages = {101-151},
title = {Bohr compactifications of discrete structures},
url = {http://eudml.org/doc/212384},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Hart, Joan
AU - Kunen, Kenneth
TI - Bohr compactifications of discrete structures
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 2
SP - 101
EP - 151
AB - We prove the following theorem: Given a⊆ω and $1 ≤ α < ω_1^{CK}$, if for some $η < ℵ_1$ and all u ∈ WO of length η, a is $Σ _α^0(u)$, then a is $Σ_α^0$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: $Σ_1^1 $-Turing-determinacy implies the existence of $0^{#}$.
LA - eng
KW - -structure; quasigroup; loop; Bohr compactifications
UR - http://eudml.org/doc/212384
ER -

References

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  7. [Mr1] D. A. Martin, The axiom of determinacy and reduction principles in the analytical hierarchy, Bull. Amer. Math. Soc. 74 (1968), 687-689. 
  8. [Mr2] D. A. Martin, Measurable cardinals and analytic games, Fund. Math. 66 (1970), 287-291. Zbl0216.01401
  9. [Ms] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980. 
  10. [Sc1] G. E. Sacks, Countable admissible ordinals and hyperdegrees, Adv. Math. 19 (1976), 213-262. Zbl0439.03027
  11. [Sc2] G. E. Sacks, Higher Recursion Theory, Springer, Berlin, 1990. 
  12. [Sm] R. L. Sami, Questions in descriptive set theory and the determinacy of infinite games, Ph.D. Dissertation, Univ. of California, Berkeley, 1976. 
  13. [Sl] J. Steel, Forcing with tagged trees, Ann. Math. Logic 15 (1978), 55-74. Zbl0404.03020
  14. [Sr] J. Stern, Evaluation du rang de Borel de certains ensembles, C. R. Acad. Sci. Paris Sér. I 286 (1978), 855-857. Zbl0377.04007

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