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**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.)

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.)

Reviewer: Jakob Kellner (Wien)

### MSC:

03E35 | Consistency and independence results |

03E17 | Cardinal characteristics of the continuum |

03E55 | Large cardinals |

### Keywords:

parametrized diamond principles; covering; bounding number; dominating number; Suslin tree; countable support iteration; Aronszajn tree; creature forcing
Full Text:
DOI

### 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 |

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[5] | Proper and improper forcing (1998) · Zbl 0889.03041 |

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[7] | DOI: 10.1007/BF02777356 · Zbl 1125.03036 |

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