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, proved in both directions. 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: intuitionistic second-order arithmetic derives each version equivalent to its principle, and four models of that theory show that no two of the six are equivalent. 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 each typed derivation, rule by rule, into a program checked by a record test that carries its own justifications. Both systems are consistent. Peano arithmetic does not prove its own consistency, while a calculus with one type universe proves it. On the line built from ratios, each familiar form of completeness has an exact price, proved in both directions, and weak König's lemma, the infinite pigeonhole principle and Ramsey's theorem for pairs are priced as well. 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; in a theory without the principle of unique choice, excluded middle cannot replace it in the completeness of the line. The final part constructs finite channels and recorders under stated preparation, control and resource assumptions, then derives limitations imposed by conservation; each theorem states its conditions and what it does not claim. Every result is proved on the page or in an appendix that its proof names. This third edition answers the public review of the second. It derives the six classification equivalences inside the theory, proves every row of the completeness table, writes out the comparison with Peano arithmetic case by case, and adds appendices that prove the consistency results. It also states the hypothesis that the second edition's sentence on excluded middle and countable choice needed: the theory has no principle of unique choice. This 391-page technical monograph contains sixteen chapters and twelve appendices, including exercise solutions. It is the mathematical companion to The Structure of Reality.