The Coding Conception of Set

Abstract

We propose the Coding Conception of ordinals and sets, which takes Cantor's three generating principles as its sole foundation. Bounded sets of ordinals are generated synchronously with the ordinals themselves through a bijective encoding function that, at each stage, selects only the finitely many bounded sets actually required by the successor, limit, and restriction principles. This selective coding yields the first-order theory $SC^{reg}$, which we establish is the metamathematically correct theory of the ordinals: it is bi-interpretable with $ZFGC^+$, yet makes no claim about the general concept of set. Extending the conception to full set theory via a monadic second-order ordinal theory with arithmetic and class comprehension produces two mutually inconsistent first-order set theories according to distinct maximality intuitions: a Type-A universe $MC_A$, in which the power set of every ordinal is a set and the universe satisfies $ZFC$; and a Type-B universe $MC_B^+$, in which sets are strictly more than ordinals and a ``largeness cardinal'' exists, beyond which power sets remain unencodable. We prove that this Power Set Dichotomy is unavoidable, even under potentialism, and conclude that $ZFC^-+$``every cardinal has a successor'' is the only philosophically uncontroversial common fragment of any true set theory; the status of the full power-set axiom remains the sole open philosophical choice point.