# Bohr compactifications of discrete structures

Fundamenta Mathematicae (1999)

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

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## Abstract

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We prove the following theorem: Given a⊆ω and $1\le \alpha <{\omega }_{1}^{CK}$, if for some $\eta <{\aleph }_{1}$ and all u ∈ WO of length η, a is ${\Sigma }_{\alpha }^{0}\left(u\right)$, then a is ${\Sigma }_{\alpha }^{0}$.We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: ${\Sigma }_{1}^{1}$-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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