# 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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topSami, 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

top- [Gn] R. O. Gandy, On a problem of Kleene's, Bull. Amer. Math. Soc. 66 (1960), 501-502. Zbl0097.24603
- [Hg] L. A. Harrington, Analytic determinacy and $0$, J. Symbolic Logic 43 (1978), 685-693.
- [Hn] J. Harrison, Recursive pseudo-well-orderings, Trans. Amer. Math. Soc. 131 (1968), 526-543. Zbl0186.01101
- [Kn] A. Kanamori, The Higher Infinite, 2nd printing, Springer, Berlin, 1997.
- [Kc] A. S. Kechris, Measure and category in effective descriptive set-theory, Ann. Math. Logic 5 (1973), 337-384. Zbl0277.02019
- [MW] R. Mansfield and G. Weitkamp, Recursive Aspects of Descriptive Set Theory, Oxford Univ. Press, Oxford, 1985. Zbl0655.03032
- [Mr1] D. A. Martin, The axiom of determinacy and reduction principles in the analytical hierarchy, Bull. Amer. Math. Soc. 74 (1968), 687-689.
- [Mr2] D. A. Martin, Measurable cardinals and analytic games, Fund. Math. 66 (1970), 287-291. Zbl0216.01401
- [Ms] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.
- [Sc1] G. E. Sacks, Countable admissible ordinals and hyperdegrees, Adv. Math. 19 (1976), 213-262. Zbl0439.03027
- [Sc2] G. E. Sacks, Higher Recursion Theory, Springer, Berlin, 1990.
- [Sm] R. L. Sami, Questions in descriptive set theory and the determinacy of infinite games, Ph.D. Dissertation, Univ. of California, Berkeley, 1976.
- [Sl] J. Steel, Forcing with tagged trees, Ann. Math. Logic 15 (1978), 55-74. Zbl0404.03020
- [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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