Abstract
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.
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Notes
and high-school algebra. Real Analysis gets its name from its focus on the properties of the so-called “real” numbers—the set of numbers that includes 0, the positive and negative whole numbers (like 12 and -7), the “rational” numbers (ratios of whole numbers, like 3/2 and -5/11), and the “irrational” numbers (like π and \( - \sqrt 2 \)). “Imaginary” numbers (like \( \sqrt { - 4} = 2i,{\text{ where }}i = \sqrt { - 1} \)) and “complex” numbers (like 3 - 5i) are the only numbers you are likely to have heard of that are not real numbers, though even they can be defined in terms of real numbers.
geometric models were constructed. As we will see in the case of Poincaré’s model, these also depend, ultimately, on the consistency of Real Analysis.
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.
Poincaré’s model consists.The formal presentation of Poincaré’s model in this paragraph, while motivated by the preceding story, is independent of the story.
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 \( \sqrt 2 \) in terms of whole numbers. See Dedekind’s essay “Continuity and Irrational Numbers” (1872; English translation, 1901), reprinted in Essays on theTheory of Numbers by Richard Dedekind (Dover, 1963).
does not use a model. See for example Chapter V of Gödel’s Proofby 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).
the system itself. From “Gödel’s Proof” by Nagel and Newman, Scientific American,June 1956.This article was later enlarged into the book mentioned in the previous note.
first decade of this century.The same that we referred to on page 10.
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Trudeau, R.J. (2001). Consistency. In: The Non-Euclidean Revolution. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2102-9_7
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