I wrote in part:
given a morphism X -> N whose pullbacks 0, 1, 2, ...: 1 -> N are all occupied [...]
Another typo; the word "along" is missing; it should be
given a morphism X -> N whose pullbacks along 0, 1, 2, ...: 1 -> N are all occupied [...]
So: Given a collection of conditions on a locally cartesian-closed category E with an initial object 0, a final object 1, and a natural-numbers object N (various refinements should be possible for more general theories), these conditions are _omega-inconsistent_ if in every such E there exists an object X and a morphism p: X -> N such that: * defining the numerals [i]: 1 -> N using the stucture maps of N (so [0]: 1 -> N, [1]: 1 -> N -> N, [2]: 1 -> N -> N -> N, etc) and letting X_i be the pullback of p: X -> N and [i]: 1 -> N, each X_i has a morphism a_i: 1 -> X_i; * letting [X,0]_N be the internal hom from X to 0 in the slice category E/N, there is (in E itself) a morphism b: 1 -> [X,0]_N. I've read this 5 times, in different orders, so there should be no mistakes. I apologise for any confusion from my abbreviations and corrections. --Toby