Concept Calculus

Harvey M. Friedman (OSU) gives a talk at the MCMP Colloquium (31 October, 2012) titled "Concept Calculus". Abstract: Concept Calculus develops theories in first and second order predicate calculus arising from the analysis of various commonsense notions. The systems arising in this way are shown to closely correspond to various well known systems arising in the foundations of mathematics - via mutual interpretations. Initial work on Concept Calculus focused on the general notions of better than and much better than. More recent developments surround a basic notion of universe, based on a principle of plenitude. Universes correspond to Peano Arithmetic. ZFC, and various extensions by large cardinals arise from the axiomatization of various basic kinds of explosions of universes. We discuss some further contexts for Concept Calculus, including the evolutionary universe, and various comparison notions. We believe that any informal conceptual context leads to the natural formulation of axiomatic principles which are mutually interpretable with a range of standard formal systems ranging from PA and fragments, through type theory and ZFC, and the usual strong systems extending ZFC.