The mathematics of the new physics developed in the Series [11-14] that we summarize here is quite extensive. However, the reconstruction of the reals without the axiom of choice is the principle focus of this paper. Combined with the enrichment of the reals by completion in the standard metric, algebraic extension and admission of a topological axiom, the new reals is established and well-defined. It is further extended to the new nonstandard analysis which, together, has the following achievements developed in the Series: (1) countably infinite counterexamples to Fermat's last theorem, disproving the conjecture; (2) solution of the problem of natural ordering of the new reals, proof of Goldbach's conjecture [13], resolution of the Banach-Tarski and Brouwers paradoxes [13]; dynamic modeling of dark matter and the superstring; and (3) characterization of undecidable propositions [14] which paved the way for the dynamic methodology. The paper also introduces the calculus of set-valued functions including the generalized integral and its application to quantum gravity. With the dynamic methodology and its main component, qualitative mathematics, as alternative to the descriptive pragmatic methodology of physics, major discoveries are achieved: (1) the solution of the turbulence, modeling and gravitational n-body problems 1 [11,20,23]; (2) discovery of ambiguous set, e.g., chaos, and proof that the real line is chaos [20, 23]; (3) development of generalized fractal [15,23,24]; algorithm for macro and quantum interactions [15,18,22] and development of the flux theory of gravitation the solution of modeling problem when the Cosmos is the given physical system. Its adaptation to Earth, is the theory of turbulence. A major achievement of qualitative mathematics, the flux theory of gravitation unites all natural forces and interactions, solves all the problems of physics and resolves its paradoxes and unanswered questions. (C) 2002 Elsevier Science Inc. All rights reserved. [References: 61]
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