Invariant Maximality and Incompleteness

We present new examples of discrete mathematical statements that can be proved from large cardinal hypotheses but not within the usual ZFC axioms for mathematics (assuming ZFC is consistent). These new statements are provably equivalent to Π1 sentences (purely universal statements, logically analogous to Fermat's Last Theorem) in particular provably equivalent to the consistency of strong set theories, including one that is in explicitly Π1 form. The examples live in the rational numbers, with only order, where the nonnegative integers are distinguished elements. The statements take the general form: every order invariant W ⊆ Q has a maximal subset S, with an invariance condition. Certain statements of this form are shown to be provably equivalent to the widely believed Con(SRP), and hence unprovable in ZFC (assuming ZFC is consistent). Modifications are made, involving a simple cross section condition, which propels the statement beyond the huge cardinal hierarchy, to attain equivalence with Con(HUGE). We also present some nondeterministic constructions of infinite and finite length with some of the same metamathematical properties. These lead to practical computer investigations designed to provide arguable confirmation of Con(ZFC) and more.