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Non-reflection of the bad set for $$\check{I}_\theta[\lambda]$$ and pcf. (English) Zbl 1324.03014
Summary: We reconsider here the following related pcf questions and make some advances:
(Q1)
concerning the ideal $$\check{I}_\kappa [\lambda]$$ how much reflection do we have for the bad set $$S^{\text{bd}}_{\lambda,\kappa} \subseteqq \{\delta < \lambda : \text{cf} (\delta) = \kappa \}$$ assuming it is well defined (for transparency only)?
(Q2)
are there somewhat free black boxes?
The advances in (Q2) will be used in subsequent for constructions of abelian groups and modules.

##### MSC:
 30000 Ordered sets and their cofinalities; pcf theory 300000 Other combinatorial set theory
##### MathOverflow Questions:
PCF theory and good points in scales
##### Keywords:
stationary sets; non-reflection; pcf
Full Text:
##### References:
 [1] Abraham, U.; Magidor, M.; Foreman, M. (ed.); Kanamori, A. (ed.), Cardinal arithmetic, (2010), Dordrecht · Zbl 1198.03053 [2] S. Spadaro, M. Kojman and D. Milovich, Order-theoretic properties of bases in topological spaces, preprint in mathematical arXive. · Zbl 1302.54008 [3] Sharon, A.; Viale, M., Some consequences of reflection on the approachability ideal, Trans. Amer. Math. Soc., 362, 4201-4212, (2010) · Zbl 1200.03029 [4] S. Shelah, Non-structure Theory, vol. accepted, Oxford University Press, Ch. I, math.LO/9201238. · Zbl 1090.03023 [5] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford University Press (1994). [6] S. Shelah, Analytical Guide and Corrections to [5] (math.LO/9906022). [7] S. Shelah, 3 lectures on pcf, paper E29 in the author’s publication list and website. [8] Shelah, S.; Baldwin, J. T. (ed.), Classification of nonelementary classes. II. abstract elementary classes, Chicago, December 1985, Berlin [9] Shelah, S., On successors of singular cardinals, Mons, 1978, Amsterdam-New York · Zbl 0449.03045 [10] Levinski, J. P.; Magidor, M.; Shelah, S., Chang’s conjecture for ℵ_{$$ω$$}, Israel J. Math., 69, 161-172, (1990) · Zbl 0696.03023 [11] Magidor, M.; Shelah, S., When does almost free imply free? (for groups, transversal etc.), J. Amer. Math. Soc., 7, 769-830, (1994) · Zbl 0819.20059 [12] Shelah, S., Reflecting stationary sets and successors of singular cardinals, Archive for Mathematical Logic, 31, 25-53, (1991) · Zbl 0742.03017 [13] Shelah, S.; Sauer, N. W. (ed.); etal., Advances in cardinal arithmetic, 355-383, (1993) · Zbl 0844.03028 [14] S. Shelah, Vive la différence III, Israel J. Math., 166 (2008), 61-96 (math.LO/0112237). · Zbl 1154.03011 [15] Dzamonja, M.; Shelah, S., On squares, outside guessing of clubs and $$I$$_{<$$f$$}[$$λ$$], Fund. Math., 148, 165-198, (1995) · Zbl 0839.03031 [16] Shelah, S., Middle diamond, Archive for Math. Logic, 44, 527-560, (2005) · Zbl 1090.03023 [17] Shelah, S., ℵ_{$$n$$}-free abelain group with no non-zero homomorphism to $$\mathbb{Z}$$, CUBO, A Mathematical Journal, 9, 59-79, (2007) [18] S. Shelah, pcf and abelian groups, Forum Mathematicum, accepted (0710.0157). · Zbl 1316.03023 [19] Goebel, R.; Shelah, S., ℵ_{$$n$$}-free modules with trivial dual, Results in Math., 54, 53-64, (2009) · Zbl 1183.13012 [20] Shelah, S., Diamonds, Proc. Amer. Math. Soc., 138, 2151-2161, (2010) · Zbl 1280.03047 [21] R. Goebel, D. Herden and S. Shelah, Prescribing endomorphism rings of ℵ_{1}-free modules, Eur. Math. Soc., accepted. [22] R. Goebel, S. Shelah and L. Struengmann, ℵ_{$$n$$}-free modules over complete discrete valuation domains with small dual, Glasgow Math. J. (2013), 12 pp. [23] S. Shelah, Quite free complicated abelian group, pcf and BB, No. 1028 in the author’s publication list and website.
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