UNIVERSAL PROPERTIES AND INCOMPLETENESS by

Every order invariant subset of Q[-r,r] has a maximal square whose sections at the x ∈ {0,...,r} agree below 0. Every universal property of the f::Q[-n,n] → Q[n,n] with no constants, has a 1-maximal regressive solution with f(0,...,n-1) ≡ f(1,...,n). We prove such purely order theoretic statements in the rationals within ZFC augmented with a standard large cardinal hypothesis, and show that ZFC does not suffice (assuming ZFC is consistent). These statements are implicitly Π1. We also establish this for explicitly Π1 forms. We refer to these developments as “Π1 mathematical incompleteness”.