# Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem

Fundamenta Mathematicae (1999)

• Volume: 160, Issue: 2, page 153-159
• 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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Sami, Ramez. "Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem." Fundamenta Mathematicae 160.2 (1999): 153-159. <http://eudml.org/doc/212385>.

@article{Sami1999,
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 = {Sami, Ramez},
journal = {Fundamenta Mathematicae},
keywords = {0 sharp; analytic determinacy; large cardinals; game-determinacy; -Turing-determinacy},
language = {eng},
number = {2},
pages = {153-159},
title = {Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem},
url = {http://eudml.org/doc/212385},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Sami, Ramez
TI - Analytic determinacy and 0# A forcing-free proof of Harrington’s theorem
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 2
SP - 153
EP - 159
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 - 0 sharp; analytic determinacy; large cardinals; game-determinacy; -Turing-determinacy
UR - http://eudml.org/doc/212385
ER -

## References

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10. [Sc1] G. E. Sacks, Countable admissible ordinals and hyperdegrees, Adv. Math. 19 (1976), 213-262. Zbl0439.03027
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