Sh:725

• Mildenberger, H., & Shelah, S. (2004). On needed reals. Israel J. Math., 141, 1–37. arXiv: math/0104276 DOI: 10.1007/BF02772209 MR: 2063023
• Abstract:
  Following Blass, we call a real a “needed” for a binary relation R on the reals if in every R-adequate set we find an element from which a is Turing computable. We show that every real needed for {\bf Cof}({\mathcal N}) is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get related notion of being “weakly needed”. We show that is is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetical reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999.
• Version 2003-05-27_11 (32p) published version (37p)

@article{Sh:725,
 author = {Mildenberger, Heike and Shelah, Saharon},
 title = {{On needed reals}},
 journal = {Israel J. Math.},
 fjournal = {Israel Journal of Mathematics},
 volume = {141},
 year = {2004},
 pages = {1--37},
 issn = {0021-2172},
 mrnumber = {2063023},
 mrclass = {03E17 (03D28 03E40)},
 doi = {10.1007/BF02772209},
 note = {\href{https://arxiv.org/abs/math/0104276}{arXiv: math/0104276}},
 arxiv_number = {math/0104276}
}