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.)

MSC:

 3e+35 Consistency and independence results 3e+17 Cardinal characteristics of the continuum 3e+55 Large cardinals
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References:

 [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
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