The iterative conception of function and the iterative conception of set
Tim Button
TL;DR
This work develops an iterative, function-centric foundation that mirrors the iterative conception of sets. By axiomatizing the iterative notion of function as Functional Level Theory ($FLT$$) and proving its synonymy with $ZF$$ via bi-interpretations, the author argues that set theory and function theory provide the same judicial foundation for mathematics, merely differing in language. The analysis extends to meta-structures (fevels, histories, and categories) and addresses consistency, coherence, and how axioms may be added extrinsically or intrinsically without breaking equivalence. A philosophical upshot is the collapse of joint-carving metaphysical programs: if the foundations are interchangeable, then privileged ontologies (membership vs application) need not be fundamental. The paper also sketches a path toward autonomous function-theoretic foundations and explores metaphysical alternatives that avoid joint-carving commitments altogether, reframing Putnam’s discussion in terms of synonymous foundational descriptions.
Abstract
Hilary Putnam once suggested that "the actual existence of sets as 'intangible objects' suffers... from a generalization of a problem first pointed out by Paul Benacerraf... are sets a kind of function or are functions a sort of set?" Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we first need to provide a full-fledged function theory. I supply such a theory: it axiomatizes the iterative notion of function in exactly the same sense that ZF axiomatizes the iterative notion of set. Indeed, this function theory is synonymous with ZF. It might seem that set theory and function theory present us with rival foundations for mathematics, since they postulate different ontologies. But appearances are deceptive. Set theory and function theory provide the very same judicial foundation for mathematics. They do not supply rival metaphysical foundations; indeed, if they supply metaphysical foundations at all, then they supply the very same metaphysical foundations.