At the beginning of Chapter 5 we argued for the mathematical legitimacy—the word we used was “consistency”—of hyperbolic geometry. (Recall that an axiomatic system is consistent if no contradiction can be deduced from its foundation of primitive terms, defined terms, and axioms.) Our case was based on two assumptions—one explicit, the other implicit.


Euclidean Geometry Axiomatic System Real Analysis Hyperbolic Geometry Circle Limit 
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  1. 3.
    Science and Hypothesis. Pages 64–68 of the 1905 English translation (Dover reprint, 1952). Poincaré describes a seemingly different but equivalent model on pages 41–43.Google Scholar
  2. 5.
    the arithmetic of the whole numbers. The other real numbers were defined in terms of the whole numbers. For example, the rational numbers were defined to be “equivalence classes” of pairs of whole numbers in which the second member was nonzero. The crucial discovery was of a method for expressing irrational numbers like √2 in terms of whole numbers. See Dedekind’s essay “Continuity and Irrational Numbers” (1872; English translation, 1901), reprinted in Essays on the Theory of Numbers by Richard Dedekind (Dover, 1963).Google Scholar
  3. 7.
    does not use a model. See for example Chapter V of Gödel’s Proof by Ernest Nagel and James R. Newman (New York University Press, 1958); or pages 31–37 of Introduction to Mathematical Logic by Elliott Mendelson (Van Nostrand, 1979).Google Scholar

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