An Equiconsistency Result on Partial Squares

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof(κ) carries a partial square. A famous theorem in set theory is the result that the failure of the square principle κ, for a regular uncountable cardinal κ, is equiconsistent with a Mahlo cardinal. Solovay proved that if λ > κ is a Mahlo cardinal, then in any generic extension by the Lévy collapse Coll(κ,<λ), λ = κ and ¬ κ. On the other hand, Jensen [6] proved that ¬ κ implies that κ is Mahlo in L. Partial square sequences were introduced by Shelah as a weakening of the square principle. Let ν < κ be regular, and let A ⊆ κ ∩ cof(ν). We say that A carries a partial square if there exists a sequence ⟨cα : α ∈ A⟩ satisfying: (a) cα is a club subset of α; (b) ot(cα) = ν; (c) if γ is a limit point of cα and cβ , then cα∩γ = cβ∩γ. A significant difference between the square principle and partial squares is that, while κ is independent of ZFC, the existence of partial squares is provable in ZFC. For example, Shelah [12] proved that if κ is a regular uncountable cardinal, then κ ∩ cof(<κ) splits into κ many pairwise disjoint subsets each of which carries a partial square. Another difference is that, unlike the square principle κ, partial squares on κ ∩ cof(κ) are consistent with κ being supercompact. For example, suppose κ is indestructibly supercompact. Then the forcing poset for adding a partial square sequence on the set κ ∩ cof(κ) with initial segments is κ-directed closed and thus preserves the supercompactness of κ. Also if V = L[E] is an extender model, then for any regular uncountable cardinal κ, κ∩cof(κ) carries a partial square. On the other hand, if κ is κ-supercompact (or even subcompact), then κ fails ([14], [3]). Magidor [8] constructed a model of set theory which satisfies a strong form of stationary set reflection, using a weakly compact cardinal. In this model there is no stationary subset of ω2 ∩ cof(ω1) which carries a partial square. In [7] we define a forcing iteration which destroys the stationarity of any subset of κ ∩ cof(κ) which carries a partial square, using a weakly compact cardinal. In this paper we show that the same forcing iteration works assuming only a greatly Mahlo cardinal. We also obtain the lower bound, by showing that if no stationary subset of κ ∩ cof(κ) carries a partial square, then κ is greatly Mahlo in L. Thus we prove the following equiconsistency result. Theorem 1. The statement that there exists a regular uncountable cardinal κ such that no stationary subset of κ ∩ cof(κ) carries a partial square is equiconsistent with a greatly Mahlo cardinal. Date: December 2010. 2010 Mathematical Subject Classification: 03E35, 03E45.