On ordinals accessible by infinitary languages

Abstract

Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if Φ is any sentence of Lλ+ω, with a unary predicate D and a binary predicate ≻, and Φ has a model M with 〈DM, ≻M〉 a well-ordering of type ≥ γ, then Φ has a model M′ where 〈DM′, ≻M′〉 is non-well-ordered. One of the interesting properties of this number is that the Hanf number of Lλ+ω is exactly δ(λ). It was proved in [BK71] that if N0 < λ < κ are regular cardinal numbers, then there is a forcing extension, preserving cofinalities, such that in the extension 2 λ = κ and δ(λ) < λ++. We improve this result by proving the following: Suppose א0 < λ < θ ≤ κ are cardinal numbers such that < θ whenever μ < θ; κλ = κ. Then there is a forcing extension preserving all cofinalities, adding no new sets of cardinality < λ, and such that in the extension 2λ = κ and δ(λ) = θ.