Geodesic
The selector principle for analytic equivalence relations does not imply the existence of an $A_2$ well ordering of the continuum
Sbornik. Mathematics, Tome 48 (1984) no. 1, pp. 159-172 
B. L. Budinas. The selector principle for analytic equivalence relations does not imply the existence of an $A_2$ well ordering of the continuum. Sbornik. Mathematics, Tome 48 (1984) no. 1, pp. 159-172. http://geodesic.mathdoc.fr/item/SM_1984_48_1_a7/
@article{SM_1984_48_1_a7,
     author = {B. L. Budinas},
     title = {The selector principle for analytic equivalence relations does not imply the existence of an $A_2$ well ordering of the continuum},
     journal = {Sbornik. Mathematics},
     pages = {159--172},
     year = {1984},
     volume = {48},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1984_48_1_a7/}
}
TY  - JOUR
AU  - B. L. Budinas
TI  - The selector principle for analytic equivalence relations does not imply the existence of an $A_2$ well ordering of the continuum
JO  - Sbornik. Mathematics
PY  - 1984
SP  - 159
EP  - 172
VL  - 48
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1984_48_1_a7/
LA  - en
ID  - SM_1984_48_1_a7
ER  - 
%0 Journal Article
%A B. L. Budinas
%T The selector principle for analytic equivalence relations does not imply the existence of an $A_2$ well ordering of the continuum
%J Sbornik. Mathematics
%D 1984
%P 159-172
%V 48
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1984_48_1_a7/
%G en
%F SM_1984_48_1_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

A set is called a selector of an equivalence relation defined on all the real numbers if it intersects each equivalence class of this relation in a singleton set. The following proposition is called the selector principle: each analytic equivalence relation on the set of all real numbers has an $A_2$-selector. It is proved that the selector principle is not equivalent to the existence of an $A_2$ well ordering of the continuum. This answers a question posed by Burgess. Equivalence is understood in the sense of equivalence in the standard Zermelo–Fraenkel set theory with the axiom of choice. Bibliography: 8 titles.

[1] Luzin N. N., Sobr. soch., T. 2, Izd-vo AN SSSR, M., 1968

[2] Burgess J., “A selection principle for $\Sigma_1^1$ equivalence relations”, Michig. Math. J., 24:1 (1977), 65–76 | DOI | MR | Zbl

[3] Mansfield R., “The nonexistence of $\Sigma_2^1$ well ordering of Cantor set”, Fund. Math., 86:3 (1975), 279–282 | MR | Zbl

[4] Kanovei V. G., “Opredelimost s pomoschyu stepenei konstruktivnosti”, Issledovaniya po teorii mnozhestv i neklassicheskim logikam, Nauka, M., 1976, 5–95 | MR

[5] Lyubetskii V. A., “Sluchainye posledovatelnosti chisel i $A_2$-mnozhestva”, Issledovaniya po teorii mnozhestv i neklassicheskim logikam, Nauka, M., 1976, 96–122

[6] Iekh T., Teoriya mnozhestv i metod forsinga, Mir, M., 1973 | MR

[7] Rodzhers X., Teoriya rekursivnykh funktsii i effektivnaya vychislimost, Mir, M., 1972 | MR

[8] Solovay R., “On the cardinality of $\Sigma_2^1$ set of reals”, Found. of Math., Springer, New-York, 1969, 58–73 | MR