The δ Foundation: Distinction, Arithmetic, and the Limits of Abstraction

Starting with a distinction that can be repeated, this book builds arithmetic and asks which conclusions follow from the construction and which require additional logical principles. Four parts develop counting, signed differences and ratios; proof rules and recognition quotients; effective descriptions, approximation and exact decision; and finite physical records. Each additional logical principle receives an exact price. With induction on every formula, Markov's rule adds nothing and the LPO rule proves exactly what excluded middle proves. The derivations that record neither LPO nor Markov's principle are Peano arithmetic, and the derivations that record no principle prove exactly the theorems of Heyting arithmetic. Six versions of the recognition classification each cost exactly one of six principles, the empty one included, and four models of intuitionistic second-order arithmetic show that no two of the six are equivalent. The strongest fragment of a typed language of iteration proves the translation of every theorem of Peano arithmetic and is conservative over it for negative sentences; the proof compiles typed derivations into programs checked by a record test that carries its own justifications. On the line built from ratios, each familiar form of completeness has an exact price. With countable choice that line is the unique solution of its task, up to exactly one isomorphism. Over intuitionistic arithmetic, countable choice proves and refutes none of LPO, WLPO, LLPO and Markov's principle, and excluded middle cannot replace it in the completeness of that line. The final part constructs finite channels and recorders under stated preparation, control and resource assumptions, then derives limitations imposed by conservation. Definitions, proofs, worked examples and exercises make the hypotheses visible at each step. This 240-page technical monograph contains sixteen chapters, seven appendices and exercise solutions. It is the mathematical companion to The Structure of Reality.