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Consistency and Independence of Axioms; Other Matters Involving Models

In: Euclidean Geometry and its Subgeometries

Author

Listed:
  • Edward John Specht

    (Indiana University South Bend)

  • Harold Trainer Jones

    (Andrews University)

  • Keith G. Calkins

    (Andrews University)

  • Donald H. Rhoads

    (Andrews University)

Abstract

The first part of this lengthy chapter shows that Cartesian (coordinate) space satisfies all thirteen of the axioms of the main development of this book. This means that the axioms are consistent since there is a model, that is, an actual mathematical system, in which all are valid. The second part constructs, for each axiom, a model in which all previously listed axioms are true, but the new one is false. This shows that the newly added axiom is independent of those previously invoked. In the third part, models are exhibited showing the mutual independence of various properties of the definitions of betweenness, mirror mappings, and reflections. The fourth part consists of models showing the insufficiency of the incidence and betweenness axioms for creation of a satisfactory geometry

Suggested Citation

  • Edward John Specht & Harold Trainer Jones & Keith G. Calkins & Donald H. Rhoads, 2015. "Consistency and Independence of Axioms; Other Matters Involving Models," Springer Books, in: Euclidean Geometry and its Subgeometries, edition 1, chapter 0, pages 413-516, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-23775-6_21
    DOI: 10.1007/978-3-319-23775-6_21
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