Pi-1-1 Normal Form in a Regular Cardinal

Abstract

A normal form is given for Π 1 1 formulas over V κ for a regular cardinal κ. Using it, a new characterization of weakly compact cardinals is given.

Sigma-1-1 Well-Founded Relations and Set Chains

“…As in [8], let L s ∈ denote L ∈ with set variables added, and let L f ∈ denote L ∈ with function variables added. Recall L f OS from Section 3.…”

“…Suppose R( H) is a relation in C ; by corollary 5 of [8] R is defined by a formula ∃ F ∀ xψ( x, F , G, H) where ψ is a ∆ 0 0 formula of L f ∈ and G are second order parameters. By theorem 9 of [9] and remarks following R( H) is C ⋪ .…”