Creatures on \(\omega_1\) and weak diamonds. (English) Zbl 1193.03075

In [Trans. Am. Math. Soc. 356, No. 6, 2281–2306 (2004; Zbl 1053.03027)], J. T. Moore, M. Hrušák and M. Džamonja introduced the parametrized \(\diamondsuit\) principles (which are compatible with the failure of CH):
For every Borel \(F:2^{<\omega_1}\to \omega^\omega\), there is a guessing sequence \(g:\omega_1\to \omega^\omega\) such that for all \(f:\omega_1\to \omega\) the real \(g(\alpha)\) “covers” the real \(F(f\restriction \alpha)\) stationary often.
Typically, the notion of “covering” is given by a cardinal characteristic \(\mathfrak x\), resulting in the parametrized \(\diamondsuit\) principle \(\diamondsuit(\mathfrak x)\). For example, \(\diamondsuit(\mathfrak b)\), which corresponds to the bounding number \(\mathfrak b\), uses the following notion of “\(r\) covers \(s\)”: There are infinitely many \(n\) such that \(r(n)>s(n)\). The dominating number \(\mathfrak d\) corresponds to: \(r(n)>s(n)\) for all but finitely many \(n\), and similarly one defines \(\diamondsuit(\text{non}(\mathcal M))\). By the Borel Galois-Tukey connections, \(\diamondsuit(\mathfrak b)\) is weaker than \(\diamondsuit(\mathfrak d)\) and also weaker than \(\diamondsuit(\text{non}(\mathcal M))\). Also, \(\diamondsuit(\text{non}(\mathcal M))\) implies the existence of a Suslin tree.
The question remained whether \(\diamondsuit(\mathfrak b)\) implies a Suslin tree as well. The paper under review gives a strong negative answer: It is consistent that even \(\diamondsuit(\mathfrak d)\) holds and that all Aronszajn trees are special.
The forcing used is a countable support iteration of creature forcings that specialize an Aronszajn tree, a modification of a construction by the author and S. Shelah [Arch. Math. Logic 42, No. 7, 627–647 (2003; Zbl 1037.03043)]: While many features of the forcing remain (such as properness, \(\omega^\omega\)-bounding, continuous reading of names), some dramatic changes are made (in particular, the new forcing does not satisfy halving) that allow the author to use an argument based on M. Hrušák’s [Acta Univ. Carol., Math. Phys. 42, No. 2, 43–58 (2001; Zbl 0999.03044)] to get \(\diamondsuit(\mathfrak d)\) in the final limit.
The creature forcing itself is very interesting; intuitively speaking it “lives” on \(\omega_1\) while the usual creature forcings live on \(\omega\). (This also applies to Zbl 1037.03043, but the paper under review is easier to read.)


03E35 Consistency and independence results
03E17 Cardinal characteristics of the continuum
03E55 Large cardinals
Full Text: DOI


[1] DOI: 10.1007/s11856-006-0005-3 · Zbl 1128.03042
[2] Acta Univ. Carotin. Math. Phys. 42 pp 43– (2001)
[3] DOI: 10.1016/0003-4843(79)90010-X · Zbl 0427.03043
[4] Surveys in set theory 8 pp 1– (1983) · Zbl 0511.00004
[5] Proper and improper forcing (1998) · Zbl 0889.03041
[6] DOI: 10.1007/s11856-007-0040-8 · Zbl 1130.03035
[7] DOI: 10.1007/BF02777356 · Zbl 1125.03036
[8] Norms on possibilities I: Forcing with trees and creatures 141 (1999) · Zbl 0940.03059
[9] Journal of Applied Analysis 3 pp 103– (1997)
[10] DOI: 10.1007/s00153-002-0168-5 · Zbl 1037.03043
[11] DOI: 10.1090/S0002-9947-03-03446-9 · Zbl 1053.03027
[12] DOI: 10.1090/S0002-9939-1987-0891159-4
[13] DOI: 10.1016/0003-4843(72)90001-0 · Zbl 0257.02035
[14] Set theory (1978)
[15] DOI: 10.1016/0168-0072(87)90082-0 · Zbl 0634.03047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.