The following results in the branches set theory, topology, nonstandard analysis, number theory, linear programming, scientific computing, Euclidean geometry and theoretical informatics are brilliant achievements! Known statements and elementary concepts such as axiom, field, etc., are as given in the relevant literature or on Wikipedia. Hence, only deviating or clarifying definitions will follow.

Unlike conventional usage in which brackets denote a more detailed explanation, bracketed parts of a statement are both optional and valid. The symbol \(\square\) finishes a proof by neglecting trivialities whereas \(\triangle\) terminates a definition. The world’s finiteness creates certain difficulties to handle the infinite. The largest exponent of a polynomial’s argument is its degree. Minimal polynomials may be specified by integer (instead of rational) coefficients as throughout in the following text.

Preliminary considerations: Referencing to philosophical terms is justified since mathematics cannot be explained in terms of metalinguistics alone and abstract words provide the most general statements according to the principle of scientificity. Here the word successive contains the physical idea that the time span between consecutive points in time is postulated, measurable or perceptible in some form. In this world, it corresponds to the Planck time of about \(10^{-43}\) seconds.

The infinite, as a dual concept to the finite (with the infinitesimal as reciprocal), can only be explained by a non-terminable process or a spatial equivalent being counterparts of the instant or the same place. This debilitates the reproach of lacking general validity by including physical facts. Due to the quantities of space and time combined in space-time, the statements below concerning space apply to time being structurally similar, too.

The abrupt transition from finite to infinite numbers, which is difficult to convey, requires mid-finite ones. The existence of actual or only potentially infinite sets remains open, since the transcendence of the infinite fails to provide a proof. From the almost “wasteful” size or expansion of the universe, it can be concluded that there is an availability that has no recognisable limits.

However, such inferences are weaker than abduction. If a finite line is broken down into an infinite number of parts, the infinite is finitely limited. If, in addition, the number of parts of both infinities is the *same*, there is mathematically an isomorphism: The enlargement of the infinitely small parts of the finite segment to finite ones results in infinity in the conventional sense in relation to the whole. That easily deduces a bijection in the mathematical sense.

Rational numbers are real, put a minimum polynomial of degree 1 to 0 and some have periodic decimal places. If the degree is \(\ge 2\), these are purely algebraic numbers with infinite numerators and denominators. Finite real fractions, together with the infinite or purely complex fractions, already form all real or complex numbers. Real continued fractions that do not terminate as finite fractions are algebraic (conventionally transcendent!) since they have an infinite denominator.

Algebra shows that sum, difference, product, and quotient of two conventional algebraic numbers of natural degree \(m\) or \(n\) are algebraic of degree at most \(mn\), and that the \(1/m\)-th power of a conventional algebraic number of degree \(n\) is also algebraic of degree at most \(mn\). If the algebraicity of a number is to be investigated and the remainder is given by the limit of a zero sequence \(\left({a}_{n}\right)\), omitting the sequence values for large \(n\) is not permitted. They are crucial.

An infinite real number can consist of a finite continued fraction with an infinitely large last denominator. If it were equated to a conventionally rational number by removing the last partial fraction, it would also be the solution to a linear equation with (infinite) whole coefficients. No (infinite) subset of complex numbers is closed. The (finite) definition offers significant advantages over axioms in terms of handling and is traditional.

Treating bijections correctly concerning the number, there are infinite many sets between the set of conventionally natural numbers and the one of conventionally real numbers. Thus, the continuum hypothesis gets a new answer. The definition of real numbers via Dedekind cuts is thus just as redundant as its definition via equivalence classes of real Cauchy sequences. The whole set theory is of course more extensive since only essential (new) ideas are presented here.

A disk without its boundary conventionally represents an open set, because then each point of it has a conventional neighbourhood that lies completely in this set. If the points are considered on a half-line, starting at the centre of the disk, there must be always a real neighbourhood for each point on this half-line towards the boundary.

This idea so far negates, however, “the end of the flagpole” whereby every such half-line must contain a point in the interior of that disk without any conventional neighbourhood there. Hence, the term openness for sets is inept^{1}cf. Heuser, Harro: *Lehrbuch der Analysis Teil 1*; 17., akt. Aufl.; 2009; Vieweg + Teubner; Wiesbaden, p. 36.. If the unit disk is considered around the origin of ordinates, so the last point of the half-line [0, 1[, dually represented, is the point \(0.\overline{1}_{2}\) and the next point is the boundary point 1. There is no other point between these two points.

For this reason, the disk without boundary is also closed, since the considered end points of the half-line just form the closure as boundary. Because the neighbourhoods do not exist on their boundary, the term closure is meaningless for sets in Euclidean space. At the same time, every open set is also closed there. This absurdity is unsettling when infinitesimal quantities are considered in a differentiated manner, thus in particular the numbers 1 (rational!) and \(0.\overline{1}_{2}\) (algebraic!) are not equated.

The absurd shows itself also by an infinite intersection of open sets such as by all open concentric disks forming a closed set, more precisely: the common centre of the disk. An infinite union of closed sets can build an open set as an open disk does, as a union of all its points as closed sets. A 0-dimensional set (point) is therefore open, because every neighbourhood also consists of one point.

Hence, the empty set \(\emptyset\) is also closed, and consequently the complete Euclidean space is closed, what can easily be generalised to higher dimensions for spheres. This special case makes also the general one absurd and meaningless: considering open or closed sets is not suitable for metric and topological spaces. Particularly, the definition of a conventional topological space appears oddly content-free and arbitrary, and becomes dubious.

Permitting infinitesimal radiuses makes the terms inner and outer point as well as boundary point meaningful, however. The conventional irrationality proof of \(\sqrt{2}\) is problematic since the square of the related rational number does not exist (of which numerator and denominator are infinite). The subject is also about conventionally unmeasurable, mid-finite and infinite sets as well as discontinuous functions. Every probabilistic statement is only valid when all its relevant possibilities were verified.

When integrating identical paths in opposite positive and negative directions, the counter-directional rule for integrals is adopted, stating that when following the path in the negative direction, the same (!) function value of both possible ones must be chosen if the function is too discontinuous, implying that the integral sums to 0 to prevent a significantly different value. Stokes’ theorem may be proven generally^{2}cf. Köhler, Günter: *Analysis*; 1. Aufl.; 2006; Heldermann; Lemgo, p. 625 f.. Function values may skip values, which are not measured then.

Conventional differentiation and integration lose the ability to distinguish between rationality and pure algebraicity in the conventional process of taking limits. This is e.g. problematic when setting about determining the roots of a polynomial exactly. Therefore, the conventional analysis cannot be preserved in its existing form and requires practicable alternatives. Period \(2\pi\) must define sine and cosine since their power series only converge for finite arguments.

The exact volume integral is the easiest to handle compared to conventional ones. Its improper form arises analogously. In some cases, suitable Landau notation may be useful. Combining function values to finitely many continuous functions, integrals may be calculated even for discontinuous functions. Here, one of both Euler-Maclaurin formulas may be used. Note that the continuity of neither integral nor derivative is assumed. An appropriate definition makes this possible.

If the result of differentiation lies outside of the domain \(D\), the closest number within the domain \(D\) should replace it. If this is not uniquely determined, the result can either be given as the set of all such numbers, or the preferred result may be selected (e.g. according to a uniform rule). The exact integral is more general than Riemann, Lebesgue(-Stieltjes) integral and other types of integral. The latter exist only in conventionally measurable sets. Extrapolations may detect bends.

Using \(1/\infty\) instead of 0 avoids a division by 0 and any vague notions of limits but requires considering carefully where this replacement makes sense and how may be exponentiated to invoke no contradiction by switching the symbols. This also allows to define integrals and differentials for each operation on real and complex numbers in such a way that every function is at least directionally integrable and differentiable wherever the function values are defined.

The definition of the exact integral via a rectangular rule may require error estimations^{3}Hämmerlin, Günther; Hoffmann, Karl-Heinz: *Numerische Mathematik*; 2. Aufl.; 1991; Springer; Berlin., ^{4}Hermann, Martin: *Numerische Mathematik*; 3., überarb. u. erw. Aufl.; 2011; Oldenbourg; München. and ^{5}Schwarz, Hans Rudolf; Köckler, Norbert: *Numerische Mathematik*; 7., überarb. Aufl.; 2009; Vieweg + Teubner; Wiesbaden.. Actual integration as the inverse operation to differentiation only makes sense for continuous functions if there is a wish to go beyond simple summation. The term predecessor, which is dual to successor, is usually not specifically mentioned again, but must be kept in mind. Only three examples illustrate the superiority of nonstandard analysis and the strength of using infinitesimal and infinite values.

Associative, commutative, and distributive laws allow to arbitrarily rearrange sums if care is taken to calculate them correctly (using Landau symbols). The Riemann series theorem is false, since the commutative law cannot be avoided, even at infinity. Arbitrariness is given if single summands are neglected or not considered at infinity. Coefficients of Taylor series may be computed very easily (for complex numbers) by using discrete Fourier transform (DFT).

DFT forms derived from this are suitable for efficient number representation and calculation. The Taylor series of the integrated logarithm, as a geometric series, allows a simple implementation of the multiplicative inverse. Functions can be determined as a vector or product of the Fourier matrix, which is a Vandermonde matrix of roots of unity, by a vector of fixed function values at the so-called crown points.

The latter are arranged in a circle around a crown centre, which should be chosen effectively and can contribute to good convergence. The product mentioned can be stored before the calculation with a freely selectable precision, which for computer calculations is best the reciprocal of a power of two. Since it can be calculated quickly and Taylor series can be realised using the Horner scheme, this method (especially as an FFT version) is very efficient.

Linear algebra achieves comparable efficiency when solving most differential equations numerically using DFT forms. In partial differential equations, the variables of dimensions other than one are fixed. Here tensors replace matrices. Absolute values are considered by using the signum function. Real numbers have usually only approximate reciprocals and can be calculated in real-time if they have sufficiently small numerators and denominators.

This book is based on ISO 80000-2:2019 (quantities and units – mathematics). The conventional notation of sums (\(\Sigma\)), products (\(\Pi\)), differentials (d and \(\partial\)), integrals (\(\smallint\)) and roots (\(\sqrt{}\)) is prevented: It is too sweeping and historically less intuitive. The minimality theorem explains the choice of 2 as basis (also for digital computers that are working most often within a binary system). In practice (of computer science), sufficiently small formal systems often help.

Simplex and numerically easy intex methods, which were developed in 35 years and answer (e.g. with the help of Gomory cuts) Hilbert’s tenth problem positively, can solve a linear programme mostly in cubic and quadratic time respectively. The simplex method allows easy and precise computing for smaller initial problems with (approximated) real fractions.

Beauty and elegance in mathematics can be ensured by adequately thinking through what is to be presented and without being stingy, reducing it to the clear essence, which justifies both and is a hallmark of the true. Unfortunately, there is a lot of ugly length in the mathematical world. It can only be hoped that this book can provide a lot of pleasure with the nonstandard mathematics and gives insight into the real good and beautiful. Who likes it, may realise both of it!

© 2002-2024 by Boris Haase

References

↑1 | cf. Heuser, Harro: Lehrbuch der Analysis Teil 1; 17., akt. Aufl.; 2009; Vieweg + Teubner; Wiesbaden, p. 36. |
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↑2 | cf. Köhler, Günter: Analysis; 1. Aufl.; 2006; Heldermann; Lemgo, p. 625 f. |

↑3 | Hämmerlin, Günther; Hoffmann, Karl-Heinz: Numerische Mathematik; 2. Aufl.; 1991; Springer; Berlin. |

↑4 | Hermann, Martin: Numerische Mathematik; 3., überarb. u. erw. Aufl.; 2011; Oldenbourg; München. |

↑5 | Schwarz, Hans Rudolf; Köckler, Norbert: Numerische Mathematik; 7., überarb. Aufl.; 2009; Vieweg + Teubner; Wiesbaden. |