No Set Carries Exactly Two Dense Linear Orders without Endpoints: A Cut-Rotation Proof in a Weak Zermelo Theory without Choice or Replacement
Abstract
Let consist of Extensionality, Pairing, Infinity, Union, Power Set, and the full Separation schema, and write when carries exactly isomorphism types of dense linear orders without endpoints. No form of Choice, Replacement, or Foundation is assumed. We prove Using countable-carrier uniqueness and the Dedekind-infinite four-type alternative from the exact-three companion, exact two forces to be neither at most countable nor Dedekind-infinite and every DLO on to be rigid. A type-count-free dyadic reflection argument eliminates the possibility that both types are self-dual, leaving one rigid non-self-dual dual pair . For each , four one-point cut orders form an antipodal two-colored square. Vertical monochromaticity would self-dualize both rays and then the whole carrier; hence . Rigidity gives a unique isomorphism . Define . Its graph is obtained by Separation inside , without forming a Replacement-generated family . Opposite-ray exchange and hereditary Dedekind-finiteness force to be an injective strictly decreasing surjection. Thus is an order reversal of , contradicting . Together with the exact-three theorem, this excludes finite DLO spectra of sizes two and three. Spectra of size at least four are not decided here.Certificates
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