@MISC{Friedman11invariantmaximal, author = {Harvey M. Friedman}, title = { INVARIANT MAXIMAL CLIQUES AND INCOMPLETENESS }, year = {2011} }

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Abstract

The Invariant Maximal Clique Theorem asserts that every graph on Q[0,n] k with a specific invariance condition has a maximal clique with another specific invariance condition. Here Q[0,n] consists of the rationals in the interval [0,n]. The invariance conditions are all given by equivalence relations involving only < on [0,n] and the distinguished elements 1,...,n. We prove the Invariant Maximal Clique Theorem using a certain well studied set theoretic hypothesis that goes well beyond the usual axioms for mathematics. The proof is modified so as to rely only on the assumption that this hypothesis is free of contradiction. We show that the Invariant Maximal Clique Theorem is, in fact, equivalent to a slight weakening of this consistency assumption.