Martin’s maximum and weak square

Abstract

Abstract. We analyse the influence of the forcing axiom Martin's Maximum on the existence of square sequences, with a focus on the weak square principle λ,μ .

P-ideal dichotomy and weak squares

“…Magidor also showed that Martin's Maximum (MM) implies the failure of κ,κ for every uncountable κ with cf(κ) = ω. Then Cummings and Magidor [1] obtained sharp results on the greatest extent of κ,κ that is compatible with MM. Magidor also considered the influence of the axiom DPFA, the analogue of PFA for proper posets that do not add any reals.…”

“…∈ Y , and hence that for any α ∈ Y , α < γ and σ α (1) is defined. Define a coloring c : (1)] because an infinite strictly decreasing sequence of ordinals can be built from this. So there are two cases to consider.…”

A microscopic approach to Souslin-tree constructions, Part I

“…Proof. The definitions of * λ and λ,ℵ1 may be found in [CM11], and the model we construct is a slight variation of the model V 3 from Section 3 of that paper. Specifically, we start by working in the model V 1 from [CM11, §3] so that κ is a supercompact cardinal indestructible under (< κ)-directed-closed notions of forcing, and CH λ holds for all cardinals λ ≥ κ.…”

Good and bad points in scales

“…We end this section by briefly remarking on two other models in which bad scales exist. A result of Magidor [5] shows that, if Martin's Maximum holds, then any scale of length ℵ ω+1 in A ℵ n , where A ⊆ ω, is bad. Foreman and Magidor, in [6], show that the same conclusion follows from the Chang's Conjecture (ℵ ω+1 , ℵ ω ) ։ (ℵ 1 , ℵ 0 ).…”