Summary: This paper presents in the fifth revised and extended edition a deeper understanding of the quantification of mathematical infinity. After eliminating the incorrect usage of bijections, key results based on the concept of cardinality become...
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Summary: This paper presents in the fifth revised and extended edition a deeper understanding of the quantification of mathematical infinity. After eliminating the incorrect usage of bijections, key results based on the concept of cardinality become obsolete, and the Poincaré conjecture must be corrected. The invalid Archimedean axiom is improved by replacing it with Archimedes' theorem. The number of algebraic numbers is counted in general and asymptotically up to a given degree, and a method for counting the elements of infinite sets is specified. The real and complex numbers are re-characterised by extending them to infinity. In the field of linear programming, a perturbation method for overcoming the problem of multiple vertices in the simplex algorithm is presented, and Smale's 9th problem is solved with the maximally semi-quintic and strongly polynomial normal algorithm.
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