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Mathematics

Mathematics
Mathematics

The following results in the branches set theory, topology, nonstandard analysis, number theory and linear programming, Euclidean geometry as well as 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 can either be included or excluded: the statement is valid in both cases. The symbol \(\square\) finishes a proof whereas \(\triangle\) terminates a definition. The world’s finiteness creates certain difficulties to handle the infinite. If numbers are zeros of a polynomial having rational coefficients, whose largest exponent of its argument is its degree, they are called algebraic, otherwise transcendental.

It may be considered whether the number of elements of the set of algebraic numbers is to define finite or infinite. Algebra teaches that the sum, difference, product, and quotient of two algebraic numbers of natural degree \(m\) or \(n\) are algebraic of degree at most \(mn\), and that the \(1/m\)-th power of an algebraic number of degree \(n\) is also algebraic of degree at most \(mn\). Transcendental numbers can be viewed as the sum of an algebraic part and a transcendental remainder.

When investigating whether a number is transcendental, if the remainder may be expressed as the limit value of a zero sequence \(\left({a}_{n}\right)\), the values of the sequence for large \(n\) may not be simply disregarded: They are important. Transcendental numbers are the numbers that lie between algebraic numbers or on either side of them. Number theory shows, if two distinct transcendental numbers (algebraic of degree \(m\)) are sufficiently close, that there is no algebraic number (of degree \(< m\)) between them.

Together with the finite rational numbers, the infinite (complex-)rational ones already numerically make up all real (complex) numbers. Therefore, algebraic and transcendental numbers are (numerically) difficult to distinguish, and approximations are of little use when determining whether a number is algebraic (to a certain degree). Real continued fractions that do not terminate as a rational number are transcendental, since they are infinite rational. They can only be used as an approximation of algebraic numbers.

Since all real numbers are approximately (infinite) rational numbers, they can be computed in real-time. Transcendental numbers allow to be satisfied with an arbitrarily precise infinitesimal number, unlike in the case of algebraic numbers, where it can be argued using the corresponding minimal polynomial or series. They therefore may be represented in the form of a rational quotient with infinite numerator and denominator. Definitions are better suited than axioms.

The finite definition offers significant advantages of handling and is traditional. It is shown that the sets of natural, integer, rational, algebraic, real or complex numbers are not closed. Hence, the difference of algebraic numbers is no longer necessarily algebraic, what complicates the theory of transcendental numbers. An infinite rational and transcendental number can also consist of a finite continued fraction. Here the last partial denominator can be infinitely big.

If it would be identified with a conventionally rational number by setting the last partial fraction equal to zero, it would simultaneously solve a linear equation with (infinite) integer coefficients. This identification leads to contradictions, if the first equation is subtracted several times from the second one and the solution of every newly emerged equation is determined. It is correct to work with approximate fractions. In practice (of computer science), sufficiently small formal systems often help.

(Conventionally natural and infinite natural) induction can show that, starting with the set of conventionally natural numbers, it can be diagonalised up to any power according to Cantor, so that all(!) infinite sets are equipotent to the set of conventionally natural numbers if Hilbert’s translations are used as an aid.

This contradictoriness is met with the theorem that there is for no set a bijection to its proper subset. Hence, Dedekind-infinity and Hilbert’s hotel are refuted, since the image sets of translations of every set lead out of the latter. Every number of elements of a set can be specified by reference to the set of conventionally natural numbers. This can be obtained only by precisely prescribing the exact construction. Therefore, \(\mathbb{N}\) is used as reference set.

\(\mathbb{N}\) and \(\mathbb{Z}\) have different many numbers. Cantor regarded such sets as of the same cardinal number. If instead bijections are correctly considered, this gives a different picture. The fuss made so far over ordinal and cardinal numbers is omitted. Associative, commutative, and distributive laws allow to arbitrarily rearrange sums if care is taken to calculate them correctly (using Landau symbols). Every probabilistic statement is only valid when all its relevant possibilities were verified.

Cantor’s distinction between merely countable and uncountable sets is not differentiated enough. The correct treatment of bijections yields the statement that, 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. Period \(2\pi\) must define sine and cosine since their power series only converge for finite arguments.

The set \(\mathbb{R}\) of all real numbers is isomorphic to a set of (hyper-) natural or integer numbers. It has both a fixed minimum element and a fixed maximum element, since \(\mathbb{R}\) is viewed holistically and completely, from which its closure is obtained together with the field axioms. Otherwise, the former theory would have to be repeatedly adapted to the circumstances. Numbers noted as sequences are plain unwieldy1cf. Landers, Dieter; Rogge, Lothar: Nichtstandard Analysis; 1. Aufl.; 1994; Springer; Berlin, p. 15 f..

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.

That was the idea before. In fact, however, “the end of the flagpole” must sometime be reached. Therefore, a point in the interior of the disk must exist, which has no conventional neighbourhood in this interior. Hence, the term openness for sets is inept2cf. 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. The former has no neighbourhood that lies in the interior of the disk, though it is an interior point. For this reason, the disk without boundary is also closed, since the last points of the half-line form, beginning from the centre of the disk, just the closure as boundary. Since 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 has not been further annoying, since infinitesimal quantities so far were not considered in a differentiated manner, thus in particular the numbers 1 and \(0.\overline{1}_{2}\) were equated. However, this is incorrect as is explained in Number Theory, since otherwise algebraicity (1) and transcendence \((0.\overline{1}_{2})\) are equated. The absurd shows itself also by an infinite intersection of open sets such as by all open concentric disks forming a closed set.

The latter is 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. These facts can easily be generalised to higher dimensions by spheres.

The terms of the inner and outer point remain meaningful however, if any infinitesimal radius is permitted whose reciprocal is infinite. Since an absurd and meaningless special case may involve the general one, considering open or closed sets is not suitable for metric and topological spaces, particularly since the definition of a conventional topological space appears oddly content-free and arbitrary.

The conventional irrationality proof of \(\sqrt{2}\) shows some dialectics and \(\mathbb{R}\) contains the multiplicative inverses usually only approximately. 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 avoid a significantly different value.

The Riemann series theorem is false for the following reasons: Summing positive summands to a desired value forces to add negative values until the original sum is attained, and vice versa. The same is true in the case that a smaller or larger value is obtained than the sum of the positive or negative terms, since the remainders almost fully cancel, and so on. Neglecting the terms shifted into infinity is irresponsible arbitrariness.

The definition of the exact integral in nonstandard analysis via a rectangular rule maybe require error estimations3Hämmerlin, Günther; Hoffmann, Karl-Heinz: Numerische Mathematik; 2. Aufl.; 1991; Springer; Berlin., 4Hermann, Martin: Numerische Mathematik; 3., überarb. u. erw. Aufl.; 2011; Oldenbourg; München. and 5Schwarz, 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. Note that the continuity of neither integral nor derivative is assumed. There may be another definition of the exact volume integral. However, the original definitions are easier to manipulate.

In some cases, suitable Landau notation may be useful. 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). Coefficients of Taylor series may be computed very easy (for complex numbers) by using discrete Fourier transform.

However, if the function values can be expressed in the form of finitely many continuous functions, integrals may be calculated even for discontinuous functions, if necessary, by appropriately applying i. a. the Euler-Maclaurin summation formula. The exact integral is more general than Riemann, Lebesgue(-Stieltjes) integral and other types of integral. The latter exist only in conventionally measurable sets. The conventional notation of sums (\(\Sigma\)), products (\(\Pi\)), differentials (d and \(\partial\)), integrals (\(\smallint\)) and roots (\(\sqrt{}\)) is prevented.

This book is based on ISO 80000-2:2019 (quantities and units – mathematics). Function values may skip values, which are not measured then. Three examples given in nonstandard analysis illustrate its superiority and the strength of using infinitesimal and infinite values. The examples of functions are only so simple for purposes of illustration. In particular, mid- and infinite sets will be considered that are conventionally non-measurable, and functions that are conventionally discontinuous.

The definition of real numbers via Dedekind cuts is thus just as unsuitable as its definition via equivalence classes of rational Cauchy sequences. Conventional differentiation and integration lose the ability to distinguish between transcendental and algebraic in the conventional process of taking limits. This is e.g. problematic, when there is a wish to determine the roots of a polynomial exactly. Therefore, the conventional analysis cannot be preserved in its existing form and requires practicable alternatives.

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 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.

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 long in the mathematical world. It is to 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!

© 2002-2020 by Boris Haase

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References

References
1 cf. Landers, Dieter; Rogge, Lothar: Nichtstandard Analysis; 1. Aufl.; 1994; Springer; Berlin, p. 15 f.
2 cf. Heuser, Harro: Lehrbuch der Analysis Teil 1; 17., akt. Aufl.; 2009; Vieweg + Teubner; Wiesbaden, p. 36.
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.