A Lifting Argument for the Generalized Grigorieff Forcing

Abstract

In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal Ä from the optimal hypothesis, while adding new unbounded subsets to Ä. In some ways these forcings are closer to the Cohen-type forcings-we show that they are not minimal-but, they share some properties with treelike forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.

Grigorieff forcing and the tree property

“…The forcing, which we now call Grigorieff forcing, was first defined by Grigorieff in [Gri71] for κ = ω; its generalizations for uncountable cardinals were studied extensively, see for example [HV16] and [AG09]. In this paper we focus on Grigorieff forcing at uncountable regular cardinals; we also mention Silver forcing at ω which has many similarities with Grigorieff forcing.…”

“…Ifḟ is a I (κ, 1)-name for a function from κ to κ + then we construct by induction a fusion sequence such that its lower bound will forceḟ is bounded. For the details for an inaccessible κ see Theorem 2.6 in [HV16]. If κ is a successor cardinal, a diamond-guided construction is usually invoked since it can show the preservation of κ + even for iterations of Grigorieff forcing (see section 2.3).…”