The spectrum of [Formula: see text]-soundness.

We study the [Formula: see text]-soundness spectra of theories. Given a recursively enumerable extension [Formula: see text] of [Formula: see text], [Formula: see text] is defined as the set of all 2-ptykes on which [Formula: see text] is correct about well foundedness. This is a measure of how close [Formula: see text] is to being [Formula: see text]-sound. We carry out a proof-theoretic classification of theories according to [Formula: see text], as well as a characterization of the sets of the form [Formula: see text]. Many of the results generalize to [Formula: see text] greater than 3. This article is part of the theme issue 'Modern perspectives in Proof Theory'.