Under the assumption of the existence of sharps for reals all simply definable posets on are classified up to forcing equivalence.
@article{bwmeta1.element.bwnjournal-article-fmv153i2p141bwm,
author = {Jind\v rich Zapletal},
title = {A classification of definable forcings on $\omega$1},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {141-144},
zbl = {0883.03038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv153i2p141bwm}
}
Zapletal, Jindřich. A classification of definable forcings on ω1. Fundamenta Mathematicae, Tome 154 (1997) pp. 141-144. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv153i2p141bwm/
[00000] [B] J. Baumgartner, Applications of the proper forcing axiom, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vaughan (eds.), North-Holland, Amsterdam, 1984, 913-959.
[00001] [BJZ] B. Balcar, T. Jech and J. Zapletal, Generalizations of Cohen algebras, Ann. Pure Appl. Logic, to appear. | Zbl 0886.03036
[00002] [J] T. Jech, Set Theory, Academic Press New York, 1978.
[00003] [JS] T. Jech and S. Shelah, Possible PCF algebras, J. Symbolic Logic 61 (1996), 313-318.
[00004] [K] S. Koppelberg, Characterization of Cohen algebras, in: Papers on General Topology and Applications, S. Andima, R. Kopperman, P. R. Misra and A. R. Todd (eds.), Ann. New York Acad. Sci. 704, 1993, 227-237. | Zbl 0830.06006
[00005] [S] S. Shelah, Proper Forcing, Springer, New York, 1982.
[00006] [T] S. Tennenbaum, Souslin's problem, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 60-63. | Zbl 0172.29503
[00007] [Z] J. Zapletal, Characterization of the club forcing, in: Papers on General Topology and Applications, S. Andima, R. C. Flagg, G. Itzkowitz, Y. Kong, R. Kopperman and P. Misra (eds.), Ann. New York Acad. Sci. 806, 1996, 476-484. | Zbl 0886.03037