The flux theory of gravitation V: the mathematics of the new physics

Applied Mathematics and Computation 130 (2002) 145–169 www.elsevier.com/locate/amc

The flux theory of gravitation V: the mathematics of the new physics E.E. Escultura

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University of the Philippines-San Fernando, 1101 Diliman, Q.C., Philippines

Abstract 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 [11,20,23];

E-mail address: [email protected] (E.E. Escultura). Present address: Blk 1, Lot 1, Granwood Villas, BF Homes, Q.C. 1120, Philippines.

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0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 0 8 8 - 1

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(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. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Flux; Chaos; Turbulence; Vortex; Fractal; Prima; Clustering; Interaction; Cosmic wave; Shock wave; Qualitative, global, physical and probabilistic mathematics

1. Introduction Until recently physics had to bear the embarrassment of hosting longstanding unsolved problems and unresolved paradoxes, e.g., gravitational nbody [11], turbulence [20,23,58] problems and paradoxes that have plagued the discipline for a long time. They include distribution of charge of the electron, stability of the protons in the nucleus despite supposedly great repulsion between them, the question of whether the Universe is finite or infinite and Einstein’s twin paradox. They are resolved in [11,12,15–23]. The resolution of problems and paradoxes came with the introduction of dynamic methodology and dynamic mathematics to overcome the inadequacy of descriptive pragmatic methodology [22]. The primary catalyst was the characterization of undecidable propositions [14]; for mathematics the principal catalyst was the discovery of loss of certainty in ambiguous sets including chaos, large and infinite sets and large and small numbers [2,23,35–38]. More recently, chaos, turbulence and fractals underscore the fact that the new physics, particularly the flux theory of gravitation, is out of range of contemporary mathematics whose foundations are the critique and rectification [14].

2. Chaos, turbulence and fractals In building a science the choice of concepts is key because it determines the appropriate tools for analysis. Even common concepts must be defined precisely to become operative. Sometimes concepts are contradictory, which could be disastrous for the discipline. For instance, chaos and turbulence are often viewed as the same phenomenon – they are not. However, turbulence generally arises from chaos and has a local chaotic component. Turbulence is also a

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special fractal and the set limit of the fractal is chaos [24,45,49]. One problem with the commonsense view of chaos as lack of order is: it rules out the use of equations for modeling it since that would be imputing order on something that does not have it; conventional mathematics offers no alternative. The more serious problem, however, is: when chaos gives rise to turbulence, a phenomenon associated with huge force, energy conservation is violated and order in the universe appears suspended in the phase of chaos. Some concepts like imaginary number ‘i’ are contradictory and do mischief to their areas of applications such as concepts of negative energy and imaginary metric. Moreover, ambiguous concepts are sources of paradoxes; for example, the use of the axiom of choice on R3 gives rise to the Banach–Tarski paradox [37, p. 269]. Definition. (1) Flux is the motion of an object or a cluster of objects with welldefined global direction, (2) turbulence is a coherence of fluxes (coherent flux), (3) chaos is a mixture of order none of which is distinguishable and (4) a fractal is an iterated sequence of any combination of translation, contraction, reflection and rotation of a set in 3-space [15] and any fractal operation that preserves the essential fractal property of self-similarity. In a broad sense, a fractal may involve distortion, replication (e.g., biological reproduction) and repetition or a multiplicity of processes or patterns; distortion vanishes at set limit. The standard dynamics considered in [20,23] has three phases: (1) phase of evolution from order to chaos, (2) transitional phase of chaos and (3) phase of evolution from chaos to order. Phases (1) and (3) have chaotic components. Standard dynamics is universal; turbulence not belonging to it is uncommon. Usual examples of this dynamics are weather and financial markets [28,57]. The standard turbulence problem is: find a mathematical model for standard dynamics that has predictive capability on its future state. The problem is solved in [20,23]; the solution, however, is undertaken at grand scale: axiomatization of physics as flux theory of gravitation [11,12,15–23] anchored on 22 fundamental physical principles including 9 biological principles [13,22]. Its adaptation to Earth is the theory of turbulence. The solution yields new technology, e.g., physical tornado breakers and technology for predicting earthquake and volcanic activity; a chemical tornado breaker is in the works. This paper summarizes and consolidates the mathematics of the theory, especially that directly related to ambiguous sets and undertakes the remaining task: put order in computation, given that the real line is chaos [23], and upgrade dynamic modeling [23] of ambiguous sets. This requires the reconstruction of real numbers (reals) without the axiom of choice or its variants and avoidance of existential and universal quantifiers on ambiguous sets.

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3. The dynamic methodology and its mathematical requirement It is difficult if not impossible for conventional mathematics to model complicated configurations of matter such as chaos, turbulence, fractals, superstrings, primal clusters and interactions and dark matter [15,18–23]. It is inadequate for modeling standard dynamics [23]. Old tools like differential equations will not do since (a) their solutions have only local validity, (b) there is no causal relationship between local and global phenomena and (c) some configurations and motion of matter are impossible to model by differential equations [16,22]. The real line, as chaos [23], is inadequate for the computational needs of the new physics. In particular, it does not have models for dark matter and the configuration of superstrings. Nor does it take into account loss of certainty in ambiguous sets. All mathematical models are descriptions of phenomena and do not establish causal relationships between the fundamental physical principles and other principles and natural phenomena. Causal relationship is the distinctive requirement of the new dynamic methodology that provides predictive capability, solves long-standing problems and resolves paradoxes and unanswered questions. The remedy: avoid deterministic mathematics including universal and existential quantifiers, the axiom of choice and the law of excluded middle, and introduce qualitative, global, physical and probabilistic mathematics capped by the reconstruction of the reals that suits computational technology and takes into account uncertainty in ambiguous sets. Qualitative mathematics in the broad sense is the complement of computational mathematics including foundations, topology, search for suitable axioms or fundamental physical principles and axiomatization of dynamic systems. A previous sequence of papers on the subject employed qualitative mathematics as a major component of the dynamic methodology. The axiomatization of physics built the flux theory of gravitation carried out in [11– 13,16–23,25]. Global mathematics in the broad sense is the complement of local mathematics, that is, mathematics in the neighborhood of a point; its main tool is differential equations. In this sense, global mathematics is part of qualitative mathematics; we give it distinctive identity since it plays a major role in the new physics. It anchors generalized fractals [7,8,15,23] and primal operations [18,22]. Physical mathematics utilizes the known solution of a dynamical system to find the solution of its mathematical model. In some cases the latter is not even necessary. For example, in the problem of breaking a tornado only qualitative and global mathematics were needed to be able to design a tornado breaker [19,20]. Physical mathematics [26] was used in the solution of the gravitational n-body problem [11]. Knowing the present structure of the galaxies and stellar and planetary systems as the stable physical solutions of prior nascent galactic, stellar and planetary systems it was necessary to work backwards to find the appropriate boundary conditions and suitable mathe-

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matical formulations to model the dynamics of systems. Aside from the qualitative and global mathematics involved in the problem the main computational mathematics involved was the Prontrjagin maximum principle [46] (updated with chattering control [6,8,59]). Probabilistic mathematics is employed in the introduction to the calculus of set-valued functions [8,18]. Among its applications are the construction of the generalized integral [12,15], calculation of kinetic energy of the primum [12,15,18] and the probabilistic solution of Fermat’s last theorem (FLT) [14]. Another feature of the dynamic methodology is the admission of fundamental physical principles, with suitable conversion principles, that are not directly observable but have verification in the visible universe. For insights, we reproduce the discussion in [18]: Suppose an anthropologist finds some footprint. He makes measurements and postulates differential equations to describe or model his observation. Then he poses these questions: (1) How heavy and large is the creature responsible for it (assuming no previous anthropological or biological information or theory is available)? (2) Is it cold blooded or warm blooded? (3) Is it erect or does it crawl? (4) How tall is it? (5) Is it well covered with hair? (6) What does it look like? (7) What level of intelligence does it possess? The equations and observation can only partially answer the first question. Knowing the softness of the ground and with a bit of calculation one can determine the weight carried by that particular foot. The envelope and contours of the footprint can be described by a suitable system of differential equations. However, all the inferences from the observation as well as the postulated differential equations do not shed any light beyond the particularities of the footprint and the nature of the foot responsible for it; they cannot even ascertain how many feet the creature has nor whether the footprint is genuine or a hoax. Definitely, the equations and observations are useless with respect to the remaining six questions. (This example highlights the inadequacy of conventional modeling.) Suppose we bring the dynamic methodology to bear on the problem by postulating the physical characteristics of the creature including the relevant physical and biological laws whose impact is verifiable. If the theory is correct suitable calculation would yield results that describe the footprint and that would constitute verification of the theory. That is not all; the theory would provide much richer information and bold predictions to guide the direction of research. If new observation disagrees with the theory, adjustment on the postulates can be made or, if disagreement is overwhelming, the theory must be thrown away and replaced by a new one. This way we develop a theory that gets better and better as more evidence comes to light. This illustrates the power of the new methodology. Its foundations are the criticisms and rectification of the descriptive pragmatic methodology summarized in [14,15, 20,22].

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Let us raise the level of complexity. Consider a very small configuration of matter, so small that it is not directly observable by present technology but its impact on visible matter is measurable (e.g., the electron belongs to dark matter but its charge is measurable). Just as in mathematics, where we take as axioms only propositions that cannot be proved from other propositions, it makes no sense to take as fundamental principle behavior of matter that is already directly verifiable. Another limitation of taking the directly verifiable as fundamental principle is: even if it has consequences referring to dark matter, they would not make sense because they are not verifiable. Therefore, we take fundamental principles about dark matter although they are motivated by known phenomena and processes in the visible universe. Most important of all, however, since we are dealing with dark matter, there should be, among the fundamental principles, suitable conversion principles that allow measurement or observation of their impact on visible matter. Then we can make the theory as rich and powerful as possible, the only requirement being that there be causal relationship between the fundamental principles and the observable phenomena and the impact of the former on visible matter is verifiable. The flux theory of gravitation that has been developed this way has much deeper and broader reach, yielding the solution of the 200-year-old gravitational nbody problem [11]. It also resolves all paradoxes and problems of infinity of quantum mechanics. In mathematics, the catalyst was FLT [14] and the dynamic methodology yields not only its resolution [14] as both undecidable and false conjecture but also the reconstruction of the reals without the axiom of choice and introduction to the new nonstandard analysis that we shall undertake as extension [14].

4. Introduction to the calculus of set-valued functions In an optimal control problem we may consider the erratic constraint equation, dx=dt ¼ gðxðtÞ; uÞdt;

ð1Þ

where u is the control parameter and dx=dt and the control function are allowed to take set-values. In fact, this is the appropriate setting for rapid oscillations [2,4,8] and toroidal fluxes of quantum gravity (see [14] for the advantages of set-valued control). We apply the technique to construct the generalized integral, one of the appropriate tools for quantum gravity [7,8,11,12,18]. Consider the topologist sine curve f ðxÞ ¼ sinn 1=x, where n is an odd positive integer. This function is set-valued at x ¼ 0, its set-value being the vertical segment ½1; 1, denoted by sinn 1=0, along the y-axis. We can, in fact, construct a function whose value at each point of an interval is the segment ½1; 1. The function ff ðxÞg ¼

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Slimsinn 1=ðs  xÞ, as s ! xþ , x 2 ½0; p, denoted by sinn 1=0x is such a function; it has the rectangular graph shown in [15, Fig. 8(a)]. We put in cross-sectional probability distribution along the vertical segment ½1; 1 at x; this distribution is uniform, that is, its expectation is constant on the interval ½p=2; p=2. It suffices to calculate the distribution at x ¼ 0. Consider the function y ¼ sinn w and its half-arc (half-cycle), the image of the interval ½p=2; p=2 under the function y. Assume that w moves from left to right at constant rate along this interval. Then the y-projection of P ðw; sinn wÞ moves upwards along vertical interval ½1; 1 on the y-axis. Near the origin, this curve oscillates rapidly and although its half-arc there is not congruent to that of sinn kw, the latter is almost congruent to it for suitably large positive integer k to any degree of approximation. Therefore, sinn kw and its derivative approximate the behavior of both the y-projection of P and that of its derivative over the interval ½p=2; p=2. We stretch that arc of sinn kw by replacing this function by sinn w and this amounts to reduction of variation of the derivative for the purpose of calculating the probability distribution. We contract it again to approximate and derive the probability distribution in accordance with the following. Mathematical Principle 20. When the maximum of the minimum horizontal distance between two simple smooth arcs with no inflection can be made arbitrarily small then both an element of arc and variation of derivative at a point of one approximate those of the other. Consider the nonoverlapping subintervals ½y; y þ dyÞ (except the uppermost subinterval of the segment which we take as ½y; 1 from this point on). The speed of the y-projection of P increases with the derivative along the corresponding subinterval of this half-arc. We ask: what is the probability that this y-projection lies in ½y; y þ dyÞ? We take, as fundamental principle for this dynamic system, the oscillation probability principle of [18], which says: The derivative dy=dx, suitably normalized, is the probability that the oscillating yprojection of P lies outside the interval ½y; y þ dyÞ. Denote dy=dx by dq=dw and by dp=dw, suitably normalized, the probability that the y-projection of P lies in this subinterval. Then we have another version of this principle: dp=dw þ dq=dw ¼ 1 or dp=dw ¼ 1  dq=dw, from which we calculate the probability distribution of the set sinn 1=0. It is clear that this probability distribution on sinn 1=0x is uniform on the x-axis, that is, the expectation is constant on x. Then the cross-sectional probability distribution of ff ðxÞg at x is dp=dwðwx Þ ¼ 2an ð1  ðn=an Þ sinn1 wx cos wx Þðan p  2Þ, where wx is a dummy variable for integration, 2an =ðan p  2Þ is the normalizing constant calculated in [18] and given by Z p=2 ð1  ðn=an Þ sinn1 w cos wÞdwx ¼ ðan p  2Þ=2an ; ð2Þ p=2

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and an is the maximum of the derivative on this half-arc, its normalizing constant, so that its value ranges from 0 to 1. This normalized probability distribution is a unit or probability measure distribution on the projection of this half-arc. The oscillation probability principle models the complementarity of speed and existence in physics expressed by Heisenberg’s uncertainty principle, which says that the faster an elementary particle travels, the less the probability that it exists in an interval along its path. We introduce the form of the generalized integral suitable for theoretical calculations, especially the calculation of the latent energy of a primum or photon. Consider the set-valued function ff ðxÞg with cross-sectional probability dp=dwðwx Þ at x. Then the generalized integral is Z bZ ws dpðws Þ; ð3Þ Q½a;b ff ðxÞg ¼ a

AðxÞ

where ws ranges over the set-value AðxÞ ¼ sinn 1=0x and dpðwx Þ ¼ ðdp=dws Þdws . The relevant set-valued function for our purposes here is cosm 1=0x ¼ Slimcosm 1=ðs  xÞ, as s ! xþ , where m is an odd positive integer. Its crosssectional probability distribution at x is uniform, that is, constant expectation and its rectangular functions cosm 1=0x and sinm 1=0x have the same probability distributions on the interval ½p=2; p=2 given by dpðwx Þ=dwx ¼ ð2am =ðam p  2ÞÞð1  ðm=am Þ sinm1 wx cos wx Þ;

ð4Þ

where the constant factors come from normalizing constants. For prima without distortions, that is, not under the influence of strong electromagnetic forces or cosmic shock waves, the value of m is 1. These rectangular functions are used to find approximate solutions of rapid product oscillations such as ðsinn 1=xÞðcosm 1=xÞ in any neighborhood of the origin which, otherwise, is not integrable by conventional integrals due to the nature of its singularity. Some functions have set-valued derivatives, among which are the infinitesimal oscillation [18], infinitesimal zigzag [59,60], and the function x2 sin 1=x2 ; the set-value of the derivative of x2 sin 1=x2 at the origin is the reals [8,18]. The normalized probability distribution of the function ðcosm 1=xÞðsinn 1=xÞ, its expectation called the generalized derivative and its generalized integral are calculated in [18]; its probability distribution is given by dpðwÞ=dw ¼ ð2  dp1 ðwÞ=dw  dp2 ðwÞ=dwÞ=a; where a¼

Z 0

tan1

pffiffiffiffiffiffiffiffi

ðn=mÞ

ð1  dq1 ðwÞ=dwÞdw þ

Z

p=2

tan1

pffiffiffiffiffiffiffiffi ð1  dq2 ðwÞ=dwÞdw: ðn=mÞ

ð5Þ

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Its generalized integral is the solution of the differential equation of the form dy=dx ¼ fgðxÞg;

ð6Þ

which may be written as Z xZ Q½a;x fgðxÞg ¼ a

ws dpðws Þ;

ð7Þ

AðxÞ

where ws ranges over AðxÞ and dpðws Þ is the cross-sectional probability distribution of fgðxÞg at s. This indefinite integral is the integral of fgðxÞg. Defining its generalized derivative as the inner integral then, for the calculus of setvalued functions, an analog of the fundamental theorem of the calculus is the equivalence of (6) and (7). The actual solution of the differential equation dy=dx ¼ ðcosm 1=xÞðsinn 1=xÞdx;

ð8Þ

where m and n are even positive integers, is computed in [18]: Z x ðcosm 1=sÞðsinn 1=sÞds; x 2 ½e; 1Þ; yðxÞ ¼ be þ e

¼ bx; ¼ bx; ¼ be þ

x 2 ½0; eÞ; Z

x 2 ðe; 0; e

ðcosm 1=sÞðsinn 1=sÞds;

x 2 ð1; e:

ð9Þ

x

This is a global solution but the generalized integral is applied only in the eneighborhood of the origin; outside, the function is Riemann integrable.

5. Applications to quantum gravity In addition to these models for quantum gravity we introduce the generalized integral for calculating the latent energy of the primum as the outer term in the nested fractal sequence of superstrings [9], i.e., the energy released when it collides with its anti-matter. We assume that it has uniform energy distribution along its cycles. Its axial projection, fLn ðxÞg ¼ a sinn bx cosm 1=0x , where n is even and m is odd, x 2 ½0; p=b, is set-valued. The envelope of the set cosm 1=0x is the pair of upper and lower half-arcs of sinusoidal curve y ¼ a sinn bx. If q is the uniform energy distribution along the cycles (in suitable units), then the latent energy of the primum is Z p=b Z EðLÞ ¼ Q½0;p=b ðfLn ðxÞgÞ ¼ a0 ðsinn bxÞws dpðws Þdx; ð10Þ 0

AðxÞ

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for suitable b, where ws ranges over AðxÞ, the set-value of sinn 1=0x . Note that sinn bx is independent of ws . The integrand of the right side of (10) represents the upper half of the primum so that its total energy is twice the value of this integral. The constant a0 absorbs this factor, the amplitude of the primum and its uniform flux density along the cycles’ axial projection. It follows from energy conservation that the energy distribution along the cycles is uniform. However, its axial projection is approximated by the curve cos m 1=x which, in turn, is approximated by the vertical segment ½0; ymax  in [18, Fig. 1(a)] and its probability distribution is integrated into dpðws Þ=dw. The inner integral is the expectation of the upper half of fLn ðxÞg. In this theory the basic constituent of visible matter, the primum, is a segment of a toroidal helix, the superstring; the rest of the superstring is in dark matter. For the first time, we have ready models for them coming from solutions of wave and diffusion equations identified in [30–33], e.g., the homogeneous harmonic wave equation, d2 u=dz2 þ k 2 u ¼ 0;

ð11Þ

simultaneously with the linear wave equation, o2 u=ot2  o2 u=ox2 þ gu ¼ 0;

ð12Þ

the Klein–Gordon equation, o2 u=ot2  Du þ dQ=du ¼ 0;

2

QðuÞ ¼ kðu2  1Þ =4

ð13Þ

and the linear Klein–Gordon–Fock wave equation, o2 u=ot2 ¼ Du  m2 u ¼ 0:

ð14Þ

Gudkov uses a suitable moving frame of reference (with the primum) to find solutions of these equations, especially (13). This is appropriate for evolutionary dynamical systems which are otherwise difficult if not impossible to solve with fixed coordinates. The prima modeled by this family of solutions are the right and left quarks, electron, positron and neutrino and their clusters like the proton and neutron, with suitable deformation. A superstring is modeled conventionally by helix u ¼ r expðikzÞ along arbitrary line z with very small amplitude r and very large frequency k. If the z-line forms a circle then the superstring forms a loop [30]. To be precise, a superstring is simply the outer loop or the first term of its fractal sequence [12,15,22]. For the harmonic wave equation (11) a moving frame of reference can be obtained by taking z ¼ x  vt and replacing g by k 2 ðv2  1Þ, where v is the velocity of the moving frame, and a is the frequency of oscillations defined by w ¼ kt, so for k 2 we find an expression k 2 ¼ w2  g. In this notation the helical solution is represented by the usual harmonic wave u ¼ r expðiðkx  wtÞÞ.

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Gerlovin [27] calculates the radii of electron’s and neutrino’s wave packets, Re ¼ 4:18323  1011 cm;

Rn ¼ 4:18219  1011 cm;

ð15Þ

as well as those of the proton’s and neutron’s, respectively (see [15]), Rp ¼ 2:20712  1014 cm;

RN ¼ 1:93201  1014 cm:

ð16Þ

Note here that the proton’s and neutron’s radii appear to be smaller than that of the electrons when it is known that the proton alone is 2000 times heavier than the electron. We interpret them to be the radii of flux tori of their constituent quarks. What we call primum here is really the wave packet of the elementary particle of quantum physics. The electron’s wave packet has radius Re of order of magnitude 1011 cm given above. Quantum physics’ electron is really its flux toroid in this paper; it has radius of order of magnitude 1016 cm [27], meaning that it is the flux toroid of a semi-agitated superstring just about at the boundary of dark and visible matter. Therefore, Rp and RN really represent the orders of magnitude of their toroidal components. Even with a moving frame of reference and clustering restriction to equatorial and polar coupling [18], it would be difficult, if not impossible, to conventionally a complicated cluster such as the large nucleus of uranium. Moreover, a conventional model does not explain the phenomena. The dynamic methodology comes to the rescue here. Consider cross-section of superstring u ¼ r expðikzÞ and its helical cylindrical segment. The order of magnitude of the radius of the cylindrical helix, that is, the radius of its semiagitated flux torus, has order of magnitude 1016 cm. Instead of solving some wave equations to provide a model, we invoke the oscillation universality and energy conservation principles to determine the basic configuration of a primum: the primum is a suitably agitated segment of the superstring that bulges as a rotation of a sinusoidal curve of the form q sinn bz (planar projection shown in [18, Fig. 3(b)]) in accordance with dark-to-visible-matter conversion principle [22]. This is the basic mathematical model of a simple primum that corresponds to the wave packet of the electron in quantum mechanics. The real number b is large and an integer (quantization principle) and yields length p=2b of the primum; h traces the helical cycle of the primum as it ranges over period 2p. The prima correspond to different values of these parameters. Any cluster of prima including the nucleus of the atom or molecules can be obtained by suitable choices of the parameters to form the basic primum followed by suitable radial deformation, translation, rotation and equatorial and polar coupling, a generalized fractal construction with generators [15]. The same approach applies to the dynamics of superconductivity and electrical conduction and transmission [22].

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6. Computational mathematics required by the new physics We undertake the reconstruction of the reals promised in [23]. However, we first summarize the rectification of foundations undertaken to resolve FLT [14]. 1. Universal language and rules of inference are avoided by making them specific to the given mathematical space since every mathematical space is defined by its own axioms; it follows that any theorem involving mapping in its proof is flawed. G€ odel’s incompleteness theorem, involving mapping from the propositional calculus to the integers, is not spared from this flaw; this is also an implicit criticism of the descriptive pragmatic methodology that relies on mathematical principles or equations for fundamental physical principles. The rectification is the axiomatization of physics. 2. The loss of certainty in large and small spaces invalidates propositions and concepts in such spaces that involve existential or universal quantifiers; this invalidates the law of excluded middle, axiom of choice and concept of limit of analysis. The remedy is avoidance of these things and admission of probabilistic mathematics, especially in the proof of undecidable propositions, in particular, probabilistic proof of FLT. The latter was found to be undecidable [14]. However, this result is apart from the question of truth of this conjecture. In fact, countably infinite counterexamples to FLT were found in [14], which established that the conjecture is false. 3. To minimize contradictions and insure applicability a physical model is required for a mathematical space. The criticisms of foundations led to the distinction between the subjective universe of thought and the objective universe of its representation, in particular, the distinction between the concept number and its numeral. One belongs to the subjective and the other to the objective universe. One’s subjective universe is not directly accessible and cannot be communicated to others with precision, nor can it be studied collectively; therefore, it cannot be axiomatized and is not the proper subject matter of mathematics. Other criticisms of foundations, especially formal logic, are carried out in [14]. The problem of ambiguity of the subjective universe, combined with involvement of the axiom of choice in the development of the reals is remarkably exhibited by several contradictions in R and its extensions including R3 [2, p. 52,35,37, p. 269,38]. Among them are the Banach–Tarski paradox and counterexample to the dichotomy axiom which establish at the same time that the irrationals are not well-defined. The latter also negates the natural ordering of R, the nested closed set theorem and, hence, the Heine–Borel theorem. In real analysis, there is a counterexample to the Lebesgue theorem on the Riemann integral [14]. These contradictions stem from attempts at algebraic and topological extension of the real line overlooking the fact that it is chaos [23]; the

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remedy is axiomatization and avoidance of deterministic mathematics on ambiguous sets including the axiom of choice and the use of the existential and universal quantifiers. In consideration of these criticisms and required foundational rectification we reconstruct R in the objective universe without the axiom of choice (or its variants including Dedekind cut and completeness axiom). The building blocks of R are the basic digits 0; 1; 2; . . . ; 9. We build the space R, þ,  consisting of terminating decimals or reals of the form X X ak ak1 . . . a1 a0 :b1 . . . bm1 bm ¼ as 10s þ bt 10t ; ð17Þ P the sum extends over s ¼ k; k  1; . . . ; 1; 0; t ¼ 1; 2; . . . ; m. The extended Hindu–Arabic numeral on the left of Eq. (17) is a decimal and the sum on the right is its size. The elements of R are rational numbers. The decimal numerals provide information about them and have their own distinctive behavior defined by their construction and subject to axioms that we shall identify. This space is uncountable. To satisfy one requirement of dynamic mathematics, we take the calculator or computer as our physical model for R, þ, . A terminating decimal is welldefined by writing a numeral – string of basic digits (left side of (17)) – and placing the decimal point in the desired place. The decimal point determines the size of the real. No uncertainty arises since only finite sets and sequences are involved so far. The numerals on the left and right of the decimal point are called integral and decimal parts, respectively. A real is an integer if its decimal part is 0. We do not allow zero as leading digit on the left unless the integral part is 0, in which case we put this digit there, before the decimal point. Unless indicated otherwise, integer refers to positive integer. The set of integers is isomorphic to the Peano naturals, under the isomorphism N ! N :000 . . . and so we may also call the integers Peano naturals in accordance with the definition in [40]. We next adopt the axioms for the additive and multiplicative operations þ and , except the inverse multiplicative axiom, of the so-called complete ordered real field F , þ,  [51, pp. 31–32]; we also leave out the completeness axiom since it involves the universal and existential quantifiers, the objectionable part of the axiom of choice on ambiguous sets. Instead we apply the enrichment methodology by introducing the standard metric on R. Then we take the completion of R in this metric and call the enriched space R , the new reals which includes limits of all Cauchy sequences in R. The standard Cauchy sequence has form N :b1 ; N :b1 b2 ; . . . ; where N is an integer and each bj is a basic digit. Each term is called a Cauchy term, obtained by adjoining a basic digit to the preceding term. A standard Cauchy sequence has a unique limit whose representation is the sequence itself; the nth term of a Cauchy sequence is

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approximation of its limit at level of accuracy 10n . When the digits of a Cauchy sequence are all 0 beyond a certain digit the limit is a terminating decimal; otherwise, it is nonterminating. Uncertainty already lurks in the latter because the tail digits of a nonterminating decimal are not known. The axiom of choice is not invoked here, however, not even in taking the limit of a Cauchy sequence, that numeral being well-defined by the sequence. To further upgrade our construction we replace the conventional notion of limit (Weierstrass sense) by the limit point of topology which allows set-limits. Uncertainty is a fact of life in ambiguous sets and cannot be eliminated. One can only minimize its impact by (1) knowing where it exists, (2) avoiding methodology tainted by it and (3) admitting suitable alternatives to deterministic proof (e.g., probabilistic proof). The terminating decimals and their additive inverses form an additive group; not so for the nonzero elements relative to multiplication. Although we have potentially new elements in this extension of R including the nonterminating decimals they are not well-defined until suitable extension is introduced. 6.1. Algebraic extension We extend the additive and multiplicative operations to R as follows: let x; y 2 R and define the map n on R by nðxÞ ¼ approximating Cauchy term of x and nðyÞ ¼ approximating Cauchy term of y (at same level of accuracy for both). Two elements of R are equal if their approximating Cauchy terms are equal at each level of approximation. Then the extended sum and product of x and y are simply written as x þ y and xy, respectively; nðx þ yÞ ¼ nðxÞ þ nðyÞ and nðxyÞ ¼ nðxÞnðyÞ at same p level p of accuracy. This is standard computation, e.g., when one computes p 2  3 he first computes the finite segment (terminating decimal) of each factor or term before proceeding with the rest of the computation. Nothing is new here except the mapping n. The lexicographic ordering is the natural ordering < on R defined as follows: two elements of R are equal if they are the same numeral. Let N :a1 a2 . . . ; M:b1 b2 . . . 2 R . Then N :a1 a2 . . . < M:b1 b2 . . . if N < M or if N ¼ M, a1 < b1 ; if a1 ¼ b1 , a2 < b2 ; . . . This is unsolved in the reals. The main achievement of the reconstruction, however, is in doing away with the axiom of choice and its ill-defining of the nonterminating decimals. The new reals is a major enrichment over the reals and extends the additive and multiplicative operations to the uncountable space beyond the terminating decimals. With this extension axiom and the standard metric on the extension space the nonterminating decimals are now well-defined. So are the additive and multiplicative operations R . We next include the standard metric on R among its axioms. Then we discover more new elements. For example, the limit of the nonstandard Cauchy sequence 0:1; 0:01; 0:001; . . . is a special nonstandard

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number of order 1=10 called dark number. We denote it by d; it satisfies d ¼ N  ðN  1Þ:999 . . . ; N ¼ 1; 2; . . . At this point we know the dark numbers of order c, 0 < c < 1; the rest of it is an open problem. There are nondenumerable new elements of R aside from the nonterminating decimals. At any rate, considering 0 and dark numbers of order k, 0 < k < 1, as an equivalence class the space R , þ, , is a complete, linearly ordered metric field. Further extension is obtained by taking the limits of divergent sequences, which enlarges R to include another kind of nonstandard numbers called unbounded numbers, the counterpart of Robinson’s infinity [50]. By taking in the decimals R inherits the structure of the decimal numerals and brings a distinctive structure on it. This space is what is presently called the real line; we call it the new real line. Most of its elements, however, including the nonterminating decimals, are not well-defined in the reals. By definition, a rational is a periodic decimal and an irrational is a nonterminating nonperiodic decimal. Multiplying a decimal number by a positive power of 10 and taking the integral part of the product yields an integer. This is one way of obtaining an integer of any magnitude. With the standard metric we may approximate any decimal by a suitable term of its standard Cauchy sequence, that is, by a decimal segment it contains. The error in this approximation is equal to the magnitude or numerical value of the tail end of the decimal expansion beyond the Cauchy approximating segment. If the decimal point is at the kth decimal place then the margin of error is 10k and this occurs when each digit after the kth decimal digit is 9. In particular, an integer is an approximation of some decimal it is an integral part of and the error is equal to the magnitude of the decimal part. When we multiply a decimal by a power of 10 and take the integral part then we reckon the margin of error in terms of the nearest power of 10 greater than or equal to that number; if that power is k, the margin of error is 10k . For example, the margin of error in approximating 781847:432 . . . by 781847 is 106 (analogous to percentage of error in physics). This completion, however, can be further extended by taking the cartesian product of R with itself, forming Cauchy sequences in this product space and bringing in the limit of all Cauchy sequences there. Since one can form a monotone expanding sequence of completion spaces completeness is relative. 6.2. The new arithmetic An important subspace of R  R is the set of twin naturals [14] of the form ðN ; N  1Þ:999 . . ., N ¼ 1; 2; . . . ; whose special element ð1; 0:999 . . .Þ is the source of the countably infinite counterexamples to FLT [14]. It is shown below that the reals of the form ðN  1Þ:999 . . . ; N ¼ 1; 2; . . . ; are also Peano naturals.

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The difference between the elements of a twin is a dark number. Since both numbers are Peano naturals this dark number is a Peano natural; this holds for the sum and product of elements of twin naturals, standard or new. The basic arithmetic of twin naturals consists of the sum and product of a natural and the nonterminating decimal element of a twin of naturals and two nonterminating decimals each belonging to a twin of naturals. Let K be a standard natural and M:999 . . . and N :999 . . . nonterminating decimals of some twins. Then 1. K þ M:999 . . . ¼ ðK þ MÞ:999 . . . ; 2. KðM:999 . . .Þ ¼ KðM þ 0:999 . . .Þ ¼ KM þ Kð0:999 . . .Þ ¼ KM þ ðK  1Þ:999 . . . ; 3. M:999 . . . þ N :999 . . . ¼ M þ N þ 0:999 . . . þ 0:999 . . . ¼ M þ N þ 2ð0:999 . . .Þ ¼ M þ N þ 1:999 . . . ; to verify that 2ð0:999 . . .Þ ¼ 1:999 . . . ; calculate ð1:999 . . .Þ=2 to obtain 0:999 . . . 4. ðM:999 . . .ÞðN :999 . . .Þ ¼ ðM þ 0:999 . . .ÞðN þ 0:999 . . .Þ ¼ MN þ Mð0:999 . . .Þ þ N ð0:999 . . .Þ þ ð0:999 . . .Þ

2

¼ MN þ ðM  1Þ:999 . . . þ ðN  1Þ:999 . . . þ 0:999 . . . ¼ MN þ ðM þ N  2Þ:999 . . . þ 0:999 . . . ¼ MN þ ðM þ N  1Þ:999 . . . ¼ ðMN þ M þ N  1Þ:999 . . .

ð18Þ

We have referred to the members of a twin as naturals. We formally established that the standard naturals, N ¼ 1; 2; . . . ; and their twins ðN  1Þ:999 . . . are both Peano naturals in accordance with [40] by establishing isomorphism between them. Consider the pair of reals fN ; ðN  1Þ:999 . . .g, N ¼ 1; 2; . . . ; where N is standard Peano natural and ðN  1Þ:999 . . . is nonterminating real. Let f be a mapping from the first element to the second element of each twin, that is, f ðN Þ ¼ ðN  1Þ:999 . . . ; f is clearly a bijection. We verify that f is an isomorphism. If N ; M are standard Peano naturals then f ðN þ MÞ ¼ ðN þ M  1Þ:999 . . . ¼ N þ M  1 þ 0:999 . . . ¼ N  1 þ M  1 þ 1:999 . . . ¼ N  1 þ 0:999 . . . þ M  1 þ 0:999 . . . ¼ ðN  1Þ:999 . . . þ ðM  1Þ:999 . . . ¼ f ðN Þ þ f ðMÞ:

ð19aÞ

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f ðNMÞ ¼ ðNM  1Þ:999 . . . ¼ NM  1 þ 0:999 . . . ¼ NM  N  M þ 1 þ N þ 1 þ M þ 1 þ 0:999 . . . ¼ NM  N  M þ 1 þ ðN  1Þ:999 . . . þ ðM  1Þ:999 . . . þ ð1Þð0:999 . . .Þ ¼ NM  N  M þ 1 þ N ð0:999 . . .Þ þ ð1Þð0:999 . . .Þ þ Mð0:999 . . .Þ þ ð1Þð0:999 . . .Þ þ 0:999 ¼ ðN  1ÞðM  1Þ þ ðN  1Þð0:999 . . .Þ þ ðM  1Þð0:999 . . .Þ 2

þ ð0:999 . . .Þ ¼ ððN  1Þ þ 0:999 . . .ÞððM  1Þ þ 0:999 . . .Þ ¼ ððN  1Þ:999 . . .ÞððM  1Þ:999 . . .Þ ¼ ðf ðN ÞÞðf ðMÞÞ:

ð19bÞ

We have shown that the standard naturals and nonterminating elements of the twin naturals are isomorphic; therefore, they are Peano naturals [40]. While the nonterminating elements are not well-defined in the reals, they are well-defined in R ; we call them new Peano naturals. Therefore, standard or new, they are Peano naturals just the same and, together with their additive inverses, form a group. Therefore, the difference between two Peano naturals is well-defined as a Peano natural. In particular, the dark number d is a Peano natural in R even if not well-defined in the reals. The first major application of the dark number in physics is the representation of the configuration of a superstring as a nested fractal sequence of superstring loop tori in which energy conservation holds [11,12]; this configuration accurately captures the notion of dissipation of energy in dark matter. We extend the basic arithmetic of nonstandard numbers. Let c be a real such that 0 < c < 1 and let d ¼ lim cn , as n ! 1, n being a natural; d is called dark number of order c. An unbounded number l of order k > 1 is defined as the limit of kn , as n ! 1. Since cn is positive and monotone decreasing d is positive and less than any positive real number. To see this, let x be any positive real; since c < 1, the natural n can be chosen large enough that 0 < d < cn < x. Similarly, it can be shown that an unbounded number of any order is greater than any real. Dark and unbounded numbers are called nonstandard numbers. The following is easy to verify: 1. The sum of a real and a dark number is a real; the sum of a real and an unbounded number is an unbounded number. 2. The product of a positive real and dark number of order c is a dark number of order c; the product of a positive real and an unbounded number of order k is an unbounded number of order k. 3. If d1 and d2 are dark numbers of orders c1 and c2 , respectively, where c1 < c2 , then d1 þ d2 is a dark number of order c2 , d2  d1 is a dark number of order

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c2 , d1 =d2 is a dark number of order c1 =c2 and d2 =d1 is an unbounded number of order c2 =c1 . 4. A positive real divided by a dark number of order c is an unbounded number of order 1=c; a positive real divided by an unbounded number of order k is a dark number of order 1=k; the reciprocal of a dark number of order c is an unbounded number of order 1=c; the reciprocal of an unbounded number of order k is a dark number of order 1=k. 5. If l1 and l2 are unbounded numbers of orders k1 and k2 , respectively, where k1 > k2 , then l1 þ l2 and l1  l2 are both unbounded numbers of order k1 and l1 =l2 and l2 =l1 are unbounded and dark numbers of orders k1 =k2 , and k2 =k1 , respectively. 6. The product of 0 and any real is 0; the sum or difference of two dark numbers of the same order is a dark number of that order; the quotient of two dark numbers or two unbounded numbers, each of the same order, is indeterminate; the difference of two unbounded numbers of the same order is also indeterminate. We have identified some nonstandard reals; their characterization is an open question. The dark numbers are models for the superstrings and the above arithmetic models operations on superstrings; the unbounded number models the timeless unbounded Universe. The nonstandard numbers are not introduced arbitrarily. They are welldefined elements of R formed with the extension and completion of R. For n example, 5  4:999 . . . ¼ limð1=10Þ , as n ! 1, is a dark number of order 1=10 (we use the standard notation for infinity and unbounded numbers because this is universally recognizable). The order of a nonstandard number is a gauge of how small or how large that number is. A dark number has nonstandard Cauchy sequence. For example, the Cauchy sequence of the dark number 5  4:999 . . . is 0:1; 0:01; 0:001; . . . ; its reciprocal, an unbounded number, does 2 3 not have a Cauchy sequence but a divergent sequence: 10; ð10Þ ; ð10Þ ; . . . For any standard natural N > 0, N  ðN  1Þ:999 . . . ¼ lim 10n , as n ! 1, a dark number. The properties of dark numbers and unbounded numbers are consistent with those of infinitesimals and infinity in Robinson’s Nonstandard Analysis [50] but we use a new terminology because the theoretical development here is different and based on a critique of Robinson’s approach. A nonstandard real is a number of the form x ¼ y þ d, where y is a standard real and d is a dark number; y and d are called the principal and minor parts, respectively. Two nonstandard numbers are equal if their principal and minor parts are equal, respectively. However, since the sum of a real and a dark number is a real, we do not represent, in actual calculation, a nonstandard number in the form g þ d, where g is real and d is a dark number, because this could lead to error; we simply apply the arithmetic of nonstandard reals to obtain g þ d ¼ g (the term dark is adapted from physics where it refers to one of

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the fundamental states of matter called dark matter that is not observable with present technology). Restricted to the reals, the extended additive and multiplicative operations coincide with conventional operations. Beyond the standard reals, i.e., terminating decimals, the arithmetic is different. For example, 0:999 þ 0:999  0:999 . . . ¼ 1:999 . . .  0:999 . . . ;

ð20Þ

which, in standard arithmetic, would have yielded 0:999 . . . ¼ 1;

ð21Þ

a contradiction. However, standard arithmetic does not apply here since 0:999 . . . is not well-defined in the reals. The correct calculation in this new arithmetic, using a technique similar to the one used for logarithms by expressing a new real as the sum of its integral and decimal parts to avoid negative decimal part, should be: 0:999 . . . þ 0:999 . . .  0:999 ¼ 1:999 . . .  0:999 . . . ¼ 0:999 . . . þ 1  0:999 . . . ¼ 0:999 . . .

ð22Þ

since 1  0:999 . . . is a dark number and is absorbed by the new real 0:999 . . . in the sum. Another astonishing theorem in the new reals is the following: Theorem. The largest and smallest reals in the open interval ð0; 1Þ of the topological real line R (with the standard metric) are 0:99 . . . and 1  0:99 . . . ; respectively. Proof. For each n, let In be an open ball of radius 102n centered at Cn . Each In lies in ð0; 1Þ. Therefore, the union of all the In lies in ð0; 1Þ and, by the second topological axiom of this metric space, this union is an open set. Since Cn lies in In for each n, then lim Cn ¼ 0:99 . . . ; as n ! 1, lies in ð0; 1Þ. To prove that 0:99 . . . is the largest real in ð0; 1Þ, let x be any point in ð0; 1Þ. Then x < 1. Since Cn is monotone increasing n can be chosen large enough that x < Cn ; let n ! 1, then x < lim Cn ¼ 0:99 . . . Therefore, 0:99 . . . is the largest real in the open interval ð0; 1Þ. To prove that 1  0:99 . . . is the smallest real in ð0; 1Þ, we note first that 1  0:99 . . . is the limit of the nonstandard Cauchy sequence 0:1; 0:01; 0:001; . . . which is monotone decreasing. Let Kn be the nth term of this sequence. For each n, let Bn be an open ball of radius 102n centered at Kn . Then each Bn lies in ð0; 1Þ and the union of the Bn is an open set and lies in ð0; 1Þ. Since, for each n, Kn lies in Bn , the limit of this sequence lies in the union of the Bn . If y is any point of ð0; 1Þ, then y > 0 and since this sequence is monotone decreasing n can be chosen large enough that y > K; let n ! 1, then y > lim Kn ¼ 1  0:99 . . . Therefore, 1  0:99 . . . is the smallest real in the open interval ð0; 1Þ. 

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Again, this theorem is not well-defined in R. The reconstruction of the real numbers either puts these astonishing results, including the calculation of ð1:99 . . .Þ=2, among the theorems of the new reals or resolves old contradictions. In the new real line every interior point of an open set has an open ball centered on it that lies entirely in the set; that open ball, however, has a dark radius. We summarize the new nonstandard analysis as extension of [14]. Consider function f ðxÞ ¼ gðxÞ þ dðxÞ in the neighborhood of real a, where x is real, gðxÞ (principal part) and dðxÞ (minor part) are real-valued functions and gðxÞ and dðxÞ tend to real and dark numbers, respectively, as x ! a. Then lim f ðxÞ ¼ limðgðxÞ þ dðxÞÞ ¼ lim gðxÞ, as x ! a. If lim gðxÞ is unbounded at a then lim f ðxÞ is unbounded, as x ! a; if lim gðxÞ is real then lim f ðxÞ ¼ lim gðxÞ, as x ! a. Calculation of limits is simplified. In an operation we treat nonstandard function f ðxÞ as binomial, the sum of principal and minor parts. In algebraic operations involving sum and product we write the result in the form F ðxÞ ¼ GðxÞ þ gðxÞ, where GðxÞ and gðxÞ are the principal and minor parts, respectively. Then lim F ðxÞ ¼ lim GðxÞ, as x ! a, where, in the calculation of the lim GðxÞ we use the intuitive approach. (In computing limits intuitively the use of a computer is admissible.) Minor parts as terms in function GðxÞ may be discarded in the calculation of its limit; if they appear as factors, the arithmetic of dark numbers applies. This intuitive approach is preferred over the defective foundations of standard calculus. Since the possible choices of c and k are nondenumerable, the nonstandard numbers form a nondenumerable subspace of R . We may also introduce the set-valued minor part of the nonstandard real. Consider the nonuniformly 2 3 4 convergent sequence S, :123; ð:312Þ ; ð:231Þ ; ð:123Þ ; . . . ; whose basis consists of cyclic permutations of digits 1, 2, 3. To find its limit we its limit we split it into three component sequences: (a) :123; :ð123Þ2 ; ð:123Þ3 ; . . . ; (b) :312; ð:312Þ2 ; ð:312Þ3 ; . . . ; (c) :231; ð:231Þ2 ; ð:231Þ3 ; . . . These sequences converge to dark numbers of orders .123, .312, .231, respectively. Therefore, the limit of S is three-valued, consisting of dark numbers of these orders. Since the number of ways of forming such component sequences is nondenumerable one may form a dark number of nondenumerable set-valued order as well as nonstandard functions with set-valued principal and minor parts. A module H ðx1 ; x2 ; . . . ; xk Þ is a rational expression in the variables x1 ; x2 ; . . . ; xk . Suppose the values of the arguments are given and H is computable as a single real, terminating or nonterminating. Then the value of H can be computed to any degree of accuracy. If this is not the case, then we do Cauchy approximations of the arguments, at consistent level of accuracy, and compute the value of H in terms of those Cauchy terms in accordance with the standard practice of computation. Given two modules H and G and using the approximation function n, we define nðH þ GÞ ¼ nðH Þ þ nðGÞ and nðHGÞ ¼ nðH ÞnðGÞ at consistent level of accuracy. In dealing with nonterminating decimals it

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may not be possible to verify equality between modules by actual computation. In this case, we say that two modules H and G are equal if and only if nðH Þ ¼ nðGÞ at every level accuracy of Cauchy approximations. Consider the function H ðxÞ ¼ hðxÞ þ dðxÞ, where hðxÞ tends to a positive real h and dðxÞ tends to a dark number d or zero, in the intuitive sense, as x ! a. Then lim H ðxÞ ¼ lim hðxÞ since the sum of a real and a dark number is real. If function rgðxÞ is small and tends to dark number of order c, c > order of dðxÞ and lim hðxÞ is a positive real then limðrgðxÞH ðxÞÞ, as x ! a, is a dark number of order c provided hðxÞ does not tend to an unbounded number. A function of the form ðrgðxÞÞhðxÞ, as x ! a, is a dark number of order a if rgðxÞ tends to the form ðaÞn , where n ! 1 as x ! a. Note that the standard or intuitive limit of H ðxÞ ¼ hðxÞ þ dðxÞ, whose principal and minor parts are hðxÞ and dðxÞ, respectively, is the principal part of its nonstandard limit. Thus, taking the standard limit of a function amounts to discarding the minor part of the nonstandard limit provided the principal part does not tend to an unbounded number. The intuitive limit is generally obtained by substitution because the limit is actually attained. For example, if f ðxÞ is the principal part of some nonstandard function, say, H ðxÞ ¼ x2 þ dðxÞ, then the standard limit of H ðxÞ, as x ! a, is a2 . The sum or product of nonstandard functions is obtained by simply considering each function as binomial, the sum of its principal and minor parts, and using the arithmetic of orders. This is also the way to handle nonstandard numbers and operations and inverse operations provided the divisor does not tend to zero. If the divisor tends to a dark number then the arithmetic of order applies. The limit of a module is obtained by simply discarding the minor parts of its terms, provided the divisor, if any, does not tend to zero. Transcendental functions and other special functions of analysis are introduced in the same manner, the only alteration being in the admission of nonstandard numbers and evaluation of limits. The arithmetic of nonstandard numbers combined with this simplified definition of limit minimizes the occurrence of indeterminate forms. For instance, if a quotient has the form d1 ðxÞ=d2 ðxÞ and the numerator and denominator tend to dark numbers of distinct positive orders c1 and c2 , respectively, their quotient c ¼ c1 =c2 is either a dark number, a new real or unbounded number depending on whether c < 1, c ¼ 1 or c > 1. Robinson’s development of nonstandard analysis parallels ours in terms of the behavior of the nonstandard numbers and approach to limits. Our departure is as follows: 1. Avoidance of the axiom of choice in the reconstruction of the reals and reduction of uncertainty to a minimum by using the basic digits 0; 1; . . . ; 9 as the building blocks. 2. Avoidance of maps in a proof as well as extension without a corresponding set of axioms; e.g., our reconstructed reals has its own set of axioms.

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3. Introduction of accurate notion of order of nonstandard reals, characterization of Robinson’s infinitesimal, namely, being positive but smaller than any real. 4. The notion of infinity is not derivable from the concept of finiteness because there is no definite boundary to cross from finite to infinite. For example, one cannot add to a finite natural another finite natural no matter how large to obtain an unbounded natural (or infinite natural in the sense of Robinson). Therefore, there is no such thing as an infinite natural obtained by the Peano successor postulate. The counterpart of Robinson’s infinity in our reconstruction is the unbounded number. 5. The notion of order provides precise evaluation of indeterminate forms. 6. The new reals is based on rectified logic and foundations. 7. The new reals without the dark numbers gives us a metric space that is not Hausdorff; with admission of open ball with dark radius, however, the space is Hausdorff. In view of (7) the only rectification of topology is admission of dark distance and, hence, open ball with dark radius. For the topological metric vector spaces, Rm , m P 1, the structure of the nonstandard reals is integrated into their topological structure. This approach to limit simplifies the calculus and nonstandard analysis considerably. 7. Counterexamples to FLT As application of the dynamic methodology we conclude this paper with the countably infinite counterexamples to FLT (with correction to [14]). Consider the standard naturals T P 1, N P 3, and the reals dn ¼ 10n , n ¼ 0; 1; 2; . . . ; x ¼ 0:999 . . .  10T , z ¼ 10T and y N ¼ lim dn , as n ! 1, where dn ¼ 0:000 . . . 01 ¼ 10n is the nth term of the nonstandard Cauchy sequence 1; 0:1; 0:01; . . . ; 0:00 . . . 01; . . . and d ¼ lim dn , as n ! 1, is a dark number. We define Cn ¼ the nth term, n ¼ 1; 2; . . . ; of the Cauchy sequence 999 . . . 9:9; 999 . . . 9:99; . . . ; 999 . . . 9:99 . . . 9; . . . of xN . Therefore, xN ¼ ð0:999 . . .Þ10NT ¼ lim Cn , as n ! 1. Define C0 ¼ integral part of xN ¼ 999 . . . 9 (string of NT 9s). Then the Cauchy sequence of xN may be written symbolically as C0 :Cn , n ¼ 1; 2; . . . Since any subsequence of a convergent sequence converges to the same limit, for c < 1 and a given N , g ¼ limðcnN Þ ¼ limðcn Þ, as n ! 1, is a dark number. Applying the above criterion for equality of modules which cannot be computed directly, we have xN þ y N ¼ zN ; ð23Þ where y ¼ d and y N ¼ dN ¼ d. To see this, we replace, for purposes of verification, the terms of Eq. (23) by their respective Cauchy terms: C0 :Cn þ dn ¼ zn ;

ð24Þ

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where zn ¼ zN ¼ 10NT . In, particular, y0 ¼ d0 ¼ 1 and (24) reduces to C0 þ 1 ¼ 9 10NT . To verify (23) for, say, T ¼ 3, N ¼ 3, h ¼ 0, we have nðxÞ ¼ 9 C0 ¼ 999 . . . 9 (string of nine 9s), nðyÞ0 ¼ y0 ¼ 1, z0 ¼ 10 and Eqs. (23) and (24) hold. The exact solution of Fermat’s equation [14] is the triple of Peano naturals ðð0:999 . . .Þ10T ; d; 10T Þ, and the solution triples of Cauchy terms for the given power N are ðCn ; 10n=N ; 10T Þ, n ¼ 0; 1; 2; . . . The case n ¼ 0 yields the integral triple ðC0 ; 1; 10T Þ satisfying Fermat’s equation. Once a solution triple ðx; y; zÞ is found, then given a standard natural k the triple ðkx; ky; kzÞ also satisfies Fermat’s equation. There are countably many such choices for k and T each of which yields a triple of Peano naturals as exact solution of Fermat’s equation and counterexample to FLT.

8. Note on the references The reader is encouraged to visit uncited references for the early rudimentary concepts data that tend to verify the new physics, e.g., on cosmic ripples and gamma ray bursts and jet outflow [1], superstrings without structure [3,44], stars revolving around a core star and planets outside our Solar System [5], introductory paper on FLT and the n-body problem [10], early concept of quantum gravity [34], early notion of black hole [29,53], conversion of superstring to molecule by brain wave [39,42], ‘‘cannibalistic’’ activity of giant galaxy [41], verification of loss of gravitational flux by a body [43], conventional models of brain waves [47,48], verification of mass of neutrino [52], verification of black hole at core of galaxy and background radiation [53,55] and galactic collision that converted superstrings to trillions of stars with pictures taken by the Hubble [54,56].

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