The discrepancy between rationals and reals was finally resolved by Eudoxus of Cnidus, a student of Plato, who reduced the comparison of irrational ratios to comparisons of multiples (rational ratios), thus anticipating Richard Dedekind's definition of real numbers.
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Later in the 19th century, the German mathematician Bernhard Riemann developed Elliptic geometry, another non-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180°.
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Wilder, Raymond L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.
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This is philosophically unsatisfying to some and has motivated additional work in set theory and other methods of formalizing the foundations of mathematics such as New Foundations by Willard Van Orman Quine.
"The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464–89.
He is a professor of mathematics at the University of Helsinki and a professor of mathematical logic and foundations of mathematics at the University of Amsterdam.
Robert Goldblatt (1984) Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics, 98).