# Definable hereditary families in the projective hierarchy

Fundamenta Mathematicae (1992)

- Volume: 140, Issue: 2, page 183-189
- ISSN: 0016-2736

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topBarua, R., and Srivatsa, V.. "Definable hereditary families in the projective hierarchy." Fundamenta Mathematicae 140.2 (1992): 183-189. <http://eudml.org/doc/211937>.

@article{Barua1992,

abstract = {We show that if ℱ is a hereditary family of subsets of $ω^ω$ satisfying certain definable conditions, then the $Δ_1^1$ reals are precisely the reals α such that $\{β:α ∈ Δ_1^1(β)\} ∉ ∈ ℱ$. This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the $Q_\{2n+1\}$-encodable reals is also shown.},

author = {Barua, R., Srivatsa, V.},

journal = {Fundamenta Mathematicae},

keywords = {projective determinacy; reals; measure; category; projective hierarchy},

language = {eng},

number = {2},

pages = {183-189},

title = {Definable hereditary families in the projective hierarchy},

url = {http://eudml.org/doc/211937},

volume = {140},

year = {1992},

}

TY - JOUR

AU - Barua, R.

AU - Srivatsa, V.

TI - Definable hereditary families in the projective hierarchy

JO - Fundamenta Mathematicae

PY - 1992

VL - 140

IS - 2

SP - 183

EP - 189

AB - We show that if ℱ is a hereditary family of subsets of $ω^ω$ satisfying certain definable conditions, then the $Δ_1^1$ reals are precisely the reals α such that ${β:α ∈ Δ_1^1(β)} ∉ ∈ ℱ$. This generalizes the results for measure and category. Appropriate generalization to the higher levels of the projective hierarchy is obtained under Projective Determinacy. Application of this result to the $Q_{2n+1}$-encodable reals is also shown.

LA - eng

KW - projective determinacy; reals; measure; category; projective hierarchy

UR - http://eudml.org/doc/211937

ER -

## References

top- [1] A. S. Kechris, Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1973), 337-384. Zbl0277.02019
- [2] A. S. Kechris, The theory of countable analytical sets, Trans. Amer. Math. Soc. 202 (1975), 259-267. Zbl0317.02082
- [3] A. S. Kechris, Effective Ramsey theorems in the projective hierarchy, in: Proceedings of the Herbrand Symposium, J. Stern (ed.), North-Holland, Amsterdam 1982. Zbl0507.03023
- [4] A. S. Kechris, D. A. Martin and R. M. Solovay, Introduction to Q-theory, in: Cabal Seminar 79-81 (Proceedings, Caltech-UCLA Logic Seminar, 1979-81), Lecture Notes in Math. 1019, Springer, 1983, 199-281.
- [5] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam 1980.
- [6] L. Piątkiewicz, A remark about separation of K-analytic sets in the product spaces, Proc. Amer. Math. Soc. 93 (1985), 363-366. Zbl0552.54028
- [7] R. M. Solovay, Hyperarithmetically encodable sets, Trans. Amer. Math. Soc. 239 (1978), 99-122. Zbl0411.03039

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