Harvey Friedman

Abstract: We show how the usual axioms for mathematics (ZFC) are insufficient even in transparent countable and finite contexts. We begin with the familiar "every countable binary relation contains a maximal square - (an A x A)". The proof is entirely constructive. We formulate "every 'nice' binary relation contains a 'nice' maximal square", using ambient spaces with modest structure. I.e., "every 'invariant' binary relation on rational [0,16]^32 contains an 'invariant' maximal square". This statement can be analyzed for purely order theoretic notions of invariance that treat 1,...,16 as distinguished. We discuss cases that can only be proved by going well beyond the usual ZFC axioms.