New Set Theory
Description
New Set Theory (NST) develops a constituent-sensitive theory of mathematical magnitude founded on atoms, hereditary support, and the Euclidean part–whole principle. Fix a foundational representa tion system ρ. Every admitted well-founded object X is assigned its representation-relative foundational support Suppρ(X), consisting of all distinct foundational atoms occurring in X at any membership depth in that representation. Its NST magnitude is CardSupp,ρ(X) := νρSuppρ(X) , where νρ is the primitive valuation associated with ρ, taking values in the ordered magnitude domain CSupp,ρ. NST sharply distinguishes logical identity, equality of foundational support, and equality of support-magnitude: X =Y, X=Supp,ρ Y ⇐⇒Suppρ(X)=Suppρ(Y), CardSupp,ρ(X) = CardSupp,ρ(Y ). Its governing Euclidean principle is Suppρ(A) ⊊ Suppρ(B) =⇒ CardSupp,ρ(A) <Supp,ρ CardSupp,ρ(B). Thus a whole is strictly greater in foundational magnitude than every object whose support is a proper part of its support, whether the supports are finite or infinite. For an object satisfying Suppρ(X) = X, one has CardSupp,ρ(X) = νρ(X). 1 This value may be denoted by |X|NST,ρ := CardSupp,ρ(X). If X is finite and Suppρ(X) = X„ finite normalization gives |X|NST,ρ = |X|C, where |X|C denotes ordinary Cantorian cardinality. For infinite sets, NST does not identify these two magnitudes. In particular, NST rejects the principle that the existence of a bijection is, by itself, suf ficient to preserve foundational magnitude. The theory combines balanced removal, total comparison, strict support inclusion, and cancellative support addition with a Full Num ber Line containing finite, infinite, and infinitesimal coordinates. Its arithmetic component is the ordered non-Archimedean field[NST] FNST postulated by Axiom 1; together with the additional finite-or-infinite