Bohr compactifications of discrete structures

Hart, Joan; Kunen, Kenneth

Fundamenta Mathematicae, Tome 159 (1999), p. 101-151 / Harvested from The Polish Digital Mathematics Library

Résumé

We prove the following theorem: Given a⊆ω and $1 \leq α < ω_{1}^{C K}$ , 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 $0^{}$ .

Publié le : 1999-01-01

Zbl 0966.54019

EUDML-ID : urn:eudml:doc:212384

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     author = {Joan Hart and Kenneth Kunen},
     title = {Bohr compactifications of discrete structures},
     journal = {Fundamenta Mathematicae},
     volume = {159},
     year = {1999},
     pages = {101-151},
     zbl = {0966.54019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv160i2p101bwm}
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Hart, Joan; Kunen, Kenneth. Bohr compactifications of discrete structures. Fundamenta Mathematicae, Tome 159 (1999) pp. 101-151. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv160i2p101bwm/

Bibliographie

[00000] [Gn] R. O. Gandy, On a problem of Kleene's, Bull. Amer. Math. Soc. 66 (1960), 501-502. | Zbl 0097.24603

[00001] [Hg] L. A. Harrington, Analytic determinacy and $0$ , J. Symbolic Logic 43 (1978), 685-693.

[00002] [Hn] J. Harrison, Recursive pseudo-well-orderings, Trans. Amer. Math. Soc. 131 (1968), 526-543. | Zbl 0186.01101

[00003] [Kn] A. Kanamori, The Higher Infinite, 2nd printing, Springer, Berlin, 1997.

[00004] [Kc] A. S. Kechris, Measure and category in effective descriptive set-theory, Ann. Math. Logic 5 (1973), 337-384. | Zbl 0277.02019

[00005] [MW] R. Mansfield and G. Weitkamp, Recursive Aspects of Descriptive Set Theory, Oxford Univ. Press, Oxford, 1985. | Zbl 0655.03032

[00006] [Mr1] D. A. Martin, The axiom of determinacy and reduction principles in the analytical hierarchy, Bull. Amer. Math. Soc. 74 (1968), 687-689.

[00007] [Mr2] D. A. Martin, Measurable cardinals and analytic games, Fund. Math. 66 (1970), 287-291. | Zbl 0216.01401

[00008] [Ms] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.

[00009] [Sc1] G. E. Sacks, Countable admissible ordinals and hyperdegrees, Adv. Math. 19 (1976), 213-262. | Zbl 0439.03027

[00010] [Sc2] G. E. Sacks, Higher Recursion Theory, Springer, Berlin, 1990.

[00011] [Sm] R. L. Sami, Questions in descriptive set theory and the determinacy of infinite games, Ph.D. Dissertation, Univ. of California, Berkeley, 1976.

[00012] [Sl] J. Steel, Forcing with tagged trees, Ann. Math. Logic 15 (1978), 55-74. | Zbl 0404.03020

[00013] [Sr] J. Stern, Evaluation du rang de Borel de certains ensembles, C. R. Acad. Sci. Paris Sér. I 286 (1978), 855-857. | Zbl 0377.04007