Gap-2 morasses of height ω

Extract

In [3] we have shown how the simplification of gap-1 morasses introduced in [4] can be extended to gap-2 morasses. As in the gap-1 case, the definition of simplified (κ, 2)-morasses (gap-2 morasses of height κ) makes sense for κ = ω, although the smallest size for Jensen's original gap-2 morasses was κ = ω₁. The existence of simplified gap-1 morasses of height ω is provable in ZFC (see [2]), but the same cannot be true for gap-2 morasses, since the top level of a simplified (ω, 2)-morass is a simplified (ω₁, 1)-morass, and the existence of simplified (ω₁, 1)-morasses is not provable in ZFC. In this paper we prove the best existence theorem for simplified (ω, 2)-morasses that we can hope for:

Theorem 1.1. There is a simplified (ω, 2)-morass iff there is a simplified (ω₁, 1)-morass.

Before giving the proof we briefly summarize the definition of simplified (ω, 2)-morasses. For a more thorough treatment, and proofs of some of the basic properties of simplified gap-2 morasses which will be used below, we refer the reader to [3]. Suppose ‹‹φ_ζ ∣ ζ ≤ ω₁›, ‹ℊ_ζξ ∣ ξ < ξ ≤ ω₁›› is a simplified (ω₁, 1)-morass. (We are using the “expanded” version of the definition.) We assume the morass is neat (see [4]). For each ζ < ω₁ let σ_ζ be the split point of ℊ_{ζ, ζ + 1}; i.e., ℊ_{ζ, ζ + 1} is a pair {id, b}, where id is the identity function, b ↾ σ_ζ = id and b(σ_ζ) = φ_ζ. A simplified (ω, 2)-morass will describe the construction of this simplified (ω, 1)-morass from finite pieces. The finite pieces will be the initial segments of the morass picked out by an increasing sequence of natural numbers ‹n_i ∣ i < ω ›. We call this piece level i of the (ω, 2)-morass, and we call the full simplified (ω₁, 1)-morass level ω. To make sure the pieces are really finite, we require that φ_x and ℊ_xy are finite for x < y < ω.