Harvey Friedman

Abstract:   According to conventional wisdom (CW), normal mathematics steers clear of foundational issues. Only a minimal fragment of the currently accepted axioms and rules for mathematics (ZFC) are used (in any remotely essential way) in current normal mathematics. The known set theoretic independence results from ZFC do not upset CW because they are known to involve abnormal subsets of uncountable sets. The known unprovability of consistency does not upset this conventional wisdom since normal mathematics is not concerned with properties of formal systems for mathematical reasoning. The study of Diophantine equations is highly normal, but the known impossibility of an algorithm does not upset CW since it does not lead to any need to reconsider the status of ZFC.

This CW has been attacked inconclusively at the margins: every Borel subset of R² that is symmetric about y=x contains or is disjoint from the graph of a Borel function. It is necessary and sufficient to use uncountably many uncountable cardinalities to prove this Theorem.

Standards are very high for the genuine overthrow of CW. The new Boolean relation theory (BRT) and its reduced forms, disjoint cover theory (DCT) and formal partition theory (FPT), promise to refute CW and ignite renewed interest in foundational issues. Initial indications are that in virtually any mathematical context (discrete or continuous), these thematic investigations are deep, open ended, varied, and explainable at the undergraduate level.

BRT grew out of two examples, which indicate its flavor. The thin set theorem asserts that for F:Z^k --> Z, there exists an infinite subset A of Z such that F[A^k] is not Z. The complementation theorem asserts that for any strictly dominating F:Z^k --> Z, there exists a (unique) infinite subset A of Z such that F[A^k]=Z\A. We present statements of this kind involving two functions and three sets provable using large cardinal axioms but not ZFC. Restricting to rather concrete functions does not change matters.

We conjecture that the general theory of such statements can be carried out with large cardinal axioms. Partial results have been obtained.