New Set Theory

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