Norms on possibilities. I: Forcing with trees and creatures.

*(English)*Zbl 0940.03059
Mem. Am. Math. Soc. 671, 167 p. (1999).

This monograph presents a very general framework for building forcing notions that add reals. When building such a forcing notion one has to balance restrictions (properties of the real) and freedom (manageable properties of the poset) in order to get something interesting. All forcing notions in the present memoir consist of conditions that look like \((w,\mathcal S)\), where \(w\) is a finite initial segment of the new real and \(\mathcal S\) limits the possible extensions of \(w\); so \((w,\mathcal S)\) is stronger than \((v,\mathcal T)\) if \(w\) is an extension of \(v\) allowed by \(\mathcal T\) and \(\mathcal S\) is some sort of combination/narrowing of the limitations given by \(\mathcal T\). An extreme case is Cohen forcing where always \(\mathcal S=\emptyset\) (complete freedom); Sacks forcing is on the opposite side: \(w\) is the root of the perfect tree \(\mathcal S\), which gives a very thin set of possible extensions indeed.

The authors present two general ways of imposing restrictions. In the first \(\mathcal S\) is a sequence \(\langle s_i\rangle_i\) of relations between finite sequences, where \(w\in\text{dom } s_0\); the simplest possible extension in this case would look like \((w',\mathcal S')\), where \((w,w')\in s_0\) and \(\mathcal S'=\langle s_i\rangle_{i\geq 1}\). A good example of such a forcing notion is in S. Shelah’s paper “On cardinal invariants of the continuum” [Axiomatic set theory, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Colo. 1983, Contemp. Math. 31, 183-207 (1984; Zbl 0583.03035)], where delicate composition methods enable one to push up the splitting number \(\mathfrak s\), while keeping the unbounding number \(\mathfrak b\) low. The second way involves trees; all well-known tree forcings suchas Laver, Miller and Sacks forcing fall into this category.

This is not a paper that one peruses ‘to get the main idea’; one needs to soak up the contents of the first two chapters to be able to read and appreciate the rest of the work. It helps to have read and studied some special cases such as the aforementioned paper [Shelah, loc. cit.], A. Blass and S. Shelah’s [“There may be simple \(P_{\alpha_1}\)- and \(P_{\alpha_2}\)-points and the Rudin-Keisler ordering may be downward directed” [Ann. Pure Appl. Log. 33, 213-243 (1987; Zbl 0634.03047)] and S. Shelah’s “Vive la différence. I: Nonisomorphism of ultrapowers of countable models” [Set theory of the continuum, Pap. Math. Sci. Res. Inst. Workshop, Berkely/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 26, 357-405 + Erratum (1992; Zbl 0789.03035)]. The rewards are considerable though; one learns to use a versatile tool for constructing forcing notions and, along the way, encounters new ones such as one that is \(\omega^\omega\)-bounding, preserves outer measure and non-meager sets but does not have the Sacks property. There is a lot to be found in this paper, the reader is invited to go exploring.

The authors present two general ways of imposing restrictions. In the first \(\mathcal S\) is a sequence \(\langle s_i\rangle_i\) of relations between finite sequences, where \(w\in\text{dom } s_0\); the simplest possible extension in this case would look like \((w',\mathcal S')\), where \((w,w')\in s_0\) and \(\mathcal S'=\langle s_i\rangle_{i\geq 1}\). A good example of such a forcing notion is in S. Shelah’s paper “On cardinal invariants of the continuum” [Axiomatic set theory, Proc. AMS-IMS-SIAM Jt. Summer Res. Conf., Boulder/Colo. 1983, Contemp. Math. 31, 183-207 (1984; Zbl 0583.03035)], where delicate composition methods enable one to push up the splitting number \(\mathfrak s\), while keeping the unbounding number \(\mathfrak b\) low. The second way involves trees; all well-known tree forcings suchas Laver, Miller and Sacks forcing fall into this category.

This is not a paper that one peruses ‘to get the main idea’; one needs to soak up the contents of the first two chapters to be able to read and appreciate the rest of the work. It helps to have read and studied some special cases such as the aforementioned paper [Shelah, loc. cit.], A. Blass and S. Shelah’s [“There may be simple \(P_{\alpha_1}\)- and \(P_{\alpha_2}\)-points and the Rudin-Keisler ordering may be downward directed” [Ann. Pure Appl. Log. 33, 213-243 (1987; Zbl 0634.03047)] and S. Shelah’s “Vive la différence. I: Nonisomorphism of ultrapowers of countable models” [Set theory of the continuum, Pap. Math. Sci. Res. Inst. Workshop, Berkely/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 26, 357-405 + Erratum (1992; Zbl 0789.03035)]. The rewards are considerable though; one learns to use a versatile tool for constructing forcing notions and, along the way, encounters new ones such as one that is \(\omega^\omega\)-bounding, preserves outer measure and non-meager sets but does not have the Sacks property. There is a lot to be found in this paper, the reader is invited to go exploring.

Reviewer: K.P.Hart (Delft)

##### MSC:

03E40 | Other aspects of forcing and Boolean-valued models |

03E05 | Other combinatorial set theory |

03E17 | Cardinal characteristics of the continuum |

03E35 | Consistency and independence results |