Namba Forcing, Weak Approximation, and Guessing

Abstract

We prove a variation of Easton's lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2 ω ≤ ω 2 is consistent with the existence of an ω 1 -distributive nowhere c.c.c. forcing poset of size ω 1 . We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle GMP follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model. Weiss[15] introduced the combinatorial principle ISP(κ), which characterizes supercompactness in the case that κ is inaccessible, but is also consistent for small values of κ such as ω 2 . The principle ISP(ω 2 ) follows from PFA, and ISP(ω 2 ) implies some of the strong consequences of PFA, such as the failure of the square principle at all uncountable cardinals. Viale-Weiss [14] introduced the idea of an ω 1 -guessing model, and proved that ISP(ω 2 ) is equivalent to the existence of stationarily many ω 1 -guessing models in P ω2 (H(θ)), for all cardinals θ ≥ ω 2 .Viale [13] proved that the singular cardinal hypothesis (SCH) follows from the existence of stationarily many ω 1 -guessing models which are also internally unbounded, which means that any countable subset of the model is covered by a countable set in the model. This raises the question of whether ISP(ω 2 ) alone implies SCH. A closely related question of Viale [13, Remark 4.3] is whether it is consistent to have ω 1 -guessing models which are not internally unbounded. Much of the work in this paper was motivated by these two questions.In this paper we introduce a weak form of ω 1 -guessing. Let κ be a regular uncountable cardinal. A model N of size ω 1 with κ ∈ N is said to be weakly κguessing if whenever f : sup(N ∩ κ) → On is a function such that for cofinally many α < sup(N ∩ κ), f ↾ α ∈ N , then there is a function g ∈ N with domain κ such that g ↾ sup(N ∩ κ) = f . We say that N is weakly guessing if N is weakly κ-guessing for all regular uncountable cardinals κ ∈ N . We will show that the existence of stationarily many weakly guessing models suffices to prove most of the strong consequences of ISP(ω 2 ), including the failure of square principles.

Guessing models imply the singular cardinal hypothesis

“…By Corollary 1.5 together with Viale's result, ISP does indeed imply SCH. 1 Corollary 1.6. ISP implies SCH.…”

“…Viale ([6, Remark 4.3]) asked whether it is consistent to have a guessing model which is not internally unbounded. In [1,Section 4] we showed that PFA implies the existence of stationarily many elementary substructures N of H(ω 2 ) of size ω 1 such that N is guessing but sup(N ∩ ω 2 ) = ω. Such models do not have countable covering, but they are internally unbounded according to Definition 1.3.…”