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#38: Alteration resp. Extension of Mathematics, Set Theory and Number Theory on 25.10.2011

The following results in the field infinity, set theory and transcendent numbers have to be called sensational!

Because of the finiteness of our world, there are certain difficulties to treat the infinite. One difficulty is the question whether one shall define the number of elements of the set of algebraic numbers finite or infinite. In the infinite case, one obtains as complement finitely transcendent numbers, in the finite case transcendent numbers. It is also shown that the set of natural, integer, rational, algebraic, real and complex numbers is not closed. Hence, the difference of (finitely) conventionally algebraic numbers is no longer necessarily (finitely) conventionally algebraic, what complicates the theory of the transcendent numbers.

A trans-rational and transcendent number can also consist of a finite continued fraction, in which the last partial denominator is infinitely large. Would it be identified with a conventionally rational number by setting the last partial fraction equal to zero, it would simultaneously solve a linear equation with integer as with trans-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, and is therefore incorrect. Correct is working with approximate fractions.

One can show by (conventionally natural and trans-natural) induction that one, starting with the set of conventionally natural numbers, can diagonalise up to any power, so that all(!) infinite sets are equipotent to the set of conventionally natural numbers, if one uses Hilbert's translations as an aid. This contradictoriness is met with the proposition that there is for no set a bijection to its proper subset. Hence, Dedekind-infinity and Hilbert's hotel are destroyed, since the image sets of translations of every set lead out of this.

Every number of elements of a set can be specified by reference to the set of conventionally natural numbers. Only from the exact construction instruction it can be specified precisely. Thus, it makes an important difference if one treats all multiples of five and constructs thereby the associated set in such a way that one times each conventionally natural number with five or that one deletes all numbers, up to each fifth, from the set of the conventionally natural numbers. Cantor regarded such sets as of the same cardinal number. If one considers, however, bijections correctly, one receives on the one hand the number of the set of the conventionally natural numbers or on the other hand a fifth of it.

Cantor's distinction between merely countable and uncountable sets is too undifferentiated. The correct treatment of bijections yields the statement that, concerning the number, there are between 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. Furthermore, the asymptotic function of the number of conventionally algebraic numbers is determined.

Since individual n-ness belongs to every natural number n that cannot be derived from its predecessors or successors, there is no complete system of axioms in mathematics, because with each new number something irreducible new emerges. If one confines, however, to selected aspects, a finite system of axioms for a finite number of entities can be specified. Each level of infinity refuses completeness all the more.

Theories are based on presuppositions. In mathematics, they are often expressed by axioms that may be true or false, what can possibly be proven by other considerations. Thus, all theories are incomplete and, as the case may be, beyond that contradictory. Instead of explicit axioms, (implicit) definitions are more suitable in that the existence of the specified is tacitly presupposed, until refutation.

The geometry gives definitions and remarks on Euclidean geometry that challenge several axioms, using results of set theory.

The question of a fair distribution of persons deals less about the theoretical content than about practical application. Here a spread-sheet analysis is used.

In the nonstandardanalysis, integration, differentiation and continuity for finite and infinite (conventionally not measurable) sets, as well as for discontinuous functions, are redefined to obtain more precise mathematical statements.

With the optimisation is presented how L solves optimisation problems (in linear or constant time by parallel processing).

Below representations simplification and exactitude of expressions are weighed against each other and the resulting desirable proceeding, deriving from this, is ethically outlined.

Below transcendent numbers necessary and sufficient criteria for (finite) transcendence are stated. There are some numbers determined as (finitely) transcendent from that one previously did not know this. (Finitely) transcendent numbers are essentially the sum of a (finitely) conventionally algebraic and a sufficiently small or sufficiently large number. The transcendence of a number can be furthermore characterised via an infinite product of conventionally complex-rational numbers and an algebraic number. (Finitely) conventionally algebraic numbers, however, are determined by the zeros of polynomials with conventionally integer or conventionally rational coefficients. From this, criteria for (finite) transcendence can be directly derived from the coefficients of the polynomials.

In the calculation of times will be shown how one can apply the octal system practically worldwide and what the advantages are here. Besides introducing a new calendar and a new calculation of clock time, the octal system is applied also to the SI-units metre (m) and second (s) and some practical examples are given. There are reasons given why the octal system can completely replace the decimal system.

Conventionally natural numbers are called the numbers 0, 1, 2, 3, ... (often also without the 0). Conventionally integer numbers are the numbers ..., -3, -2, -1, 0, 1, 2, 3, ... Conventionally rational numbers are represented by fractions with conventionally integer numerator and conventionally natural denominator unequal 0.

Sufficiently known axioms define more precisely, beyond that, the conventionally real and complex numbers, which can be continued in all eternity and is realised this way in the following, where the completeness axiom can be relinquished. The set of conventionally natural numbers without 0 is denoted with ℕ (including 0 with ℕ0, this also for other sets), the set of conventionally integer numbers with ℤ, the set of conventionally rational numbers with ℚ, the set of conventionally real numbers with ℝ and the set of conventionally complex numbers with ℂ.

Definition: A natural number is called finitely generated, if it emerged from finitely many additions from 1 to 0. A complex number whose absolute value is greater than any finitely generated natural number is called infinite. A complex number different from zero whose absolute value of its reciprocal is infinite is called infinitesimal. A complex number that is neither infinite nor infinitesimal is called properly finite. Properly finite and infinitesimal numbers are called finite. Conventionally real numbers that are not conventionally rational are called irrational.

The sets continued into infinity are preceded by ~, numbers residing in infinity by trans- and their corresponding sets by T. The conventional notation is used for the conventional and trans-conventional numbers. If the conventional numbers are meant, this is indicated by the preceding word conventional. If a subset of a set M consists of all positive (negative) elements of M, so it is also denoted with M+ (M-). The number of all elements of a set M is denoted with |M|. If the carrier of an interval is unclear, the former follows down the latter.

Three sets lemma: A maximal absolute value of all properly finite numbers exists just as little as a minimal (maximal) absolute value of all infinitesimal or infinite numbers. Thus, there exists in each case also no set of all numbers of a mentioned type.

Proof: Assuming the opposite, in the remaining cases, another infinitesimal or infinite (resp. properly infinite) number can be generated by multiplying the minimal (maximal) absolute value with ½ (2), what contradicts the minimality (maximality) of the absolute value.

The assertion that |ℝ| should equal 2|ℕ| states that all conventionally real numbers can be developed into dual numbers. Against this speaks, on the one hand, that many conventionally rational numbers can be represented only approximately as dual numbers and much more that conventionally real numbers may be also sums with smaller exponents of 2 than the 2-n set minimal. For example, the number ⅓ is only approximately included as 0.01010101... and must be distinguished from it.

Proposition: For no set, there is a bijection to its proper subset.

Proof: We prove this by (conventionally natural and trans-natural) induction, starting with the singleton and progressing over multi-element sets by successively adding an element. The same result is obtained if one removes an element from a set and wants to find a bijection for the resulting set: There is none, because the missing element cannot be replaced. The natural induction again proves the claim.

Conclusion: Thus especially Dedekind-infinity and Cantor's diagonal argument are discounted, since ℕ is a proper subset of ℚ. A translation of an infinite set always leads out of this set. Thus Hilbert's hotel is destroyed. Concerning the continuum hypothesis is to say that there is an infinite number of sets whose number of elements lies between that of ℕ and that of ℝ.

Completion lemma: The set of (conventionally) natural numbers is not closed concerning addition.

Proof: If one forms by bijection to (~)ℕ the set 2(~)ℕ, so this contains elements that cannot be contained in (~)ℕ, since, for example, 1, 3, 5 are missing in 2(~)ℕ. On the other hand, for all k ∈ (~)ℕ also k + k must be included in (~)ℕ. This contradiction implies the assertion.

Conclusion: The (conventionally) rational, algebraic, real and complex numbers are also not closed concerning addition, because they are based on the addition of (conventionally) natural numbers. The lack of completion is also transferable to multiplication and inversion, since they build on the addition of natural numbers. All this results especially from the following

Minimax proposition: In every non-empty partially ordered set, each chain contains exactly one minimal and one maximal element, and thus the set itself at least one each.

Proof: For finite sets, the claim is clear. Infinite chains are isomorphic to sets consisting only of consecutive ordinal numbers - starting with 0. Consider now the set {0, 1, 2, ... , 2|ℕ|}. This is isomorphic to the homogeneous set that consists of the numbers {0, 1, ½, ¼, ¾, ⅛, ⅜, ⅝, ⅞, ...}, emerged by successively bisecting the distances of the elements, by dint of the mapping k ↦ k / 2|ℕ|. It has exactly the minimal element 0 and the maximal element 1. By thinning, the assertion holds also for the set ℕ by dint of the mapping 2k ↦ k, with what the notation |ℕ|, with the maximum element |ℕ|, is justified. The assertion now follows by natural induction for all smaller ordinal number sets by removing elements from {0, 1, ½, ¼, ¾, ⅛, ⅜, ⅝, ⅞, ...} with 1/2|ℕ|-steps, and for all bigger ordinal number sets by successively still increasing the exponent |ℕ|.

Indefiniteness lemma: About the minimal and maximal elements of an infinite set only a statement can be made if there are decision criteria present. This particularly applies to the element with the smallest or the greatest absolute value of the real and complex numbers.

Proof: Clear.

Example: It is undecidable whether |ℕ| is even or not, since who examines the middle of ℕ as homogeneous isomorphic interval [0, 1] can only find something (extensive) and not nothing, which would be, contradictorily, divisible, what makes the argument circular. Therefore we cannot say, whether being or non-being is attributed to the middle, whether it is element or not. Therewith ℕ is also not countable and |ℕ| mod n can be determined for no finite n ∈ ℕ \ {1}.

Definition: Let the basic set ℕ, on which all other infinite sets can be built, be on this homepage always the one of the elements that emerge if one adds successively 1 to 0 with the same operator for the addition.

Remark: Therewith, ℕ is uniquely determined, but not in every respect. From ℕ = {1} ∪ {n + 1 : n ∈ ℕ} and 1 + max ℕ ∉ ℕ, no contradiction can be construed, since from the assumption ∃ n ∈ ℕ (n + 1 ∉ ℕ) only follows, without contradiction, n = max ℕ. In all subsets of ~ℂ, the operators permitted in all subsets of ℂ can be used in the same way (transfer principle), which is to be shown in individual cases, but remains undone here because of their number.

Definition: Let Euler's number be (symbolically) defined by the equation e = max (1 + 1/g)g < ℜ := max ℝ with trans-natural g and it shall (approximately) apply ex+y = ex ey for all complex x and y. Let the mathematical constant π be (symbolically) defined by the equation e = -1, with i as the imaginary unit. Then, let a logarithm function log be (symbolically) defined by elog z = z and the corresponding power function by es log z with complex s and z. In this way, the exponentiation can be (symbolically) defined. Computationally, one will usually have to be content with approximations.

Lemma: The Archimedean property does not apply for infinitely many real numbers.

Proof: Let a ∈ ℝ+ \ ]0, 1] and b = 1/|ℕ|. Then applies b n ≤ 1 < a for all n ∈ ℕ.

Archimedean proposition: There is a (finite) n ∈ ℕ with b n > a, iff for a > b with a, b ∈ ℝ+ a/b < |ℕ| (resp. a/b is finite).

Proof: If a/b ≥ |ℕ| applies, then also a/b ≥ n is valid for all n ∈ ℕ. Iff a/b can be rounded up to the next greatest finite natural number, the claim in parentheses is valid. If a/b is finite, this is always possible, since adding 1 to a finite number, as finite operation, is also always possible and yields a finite sum.

Definition: The sum p(x) = amxm + am-1xm-1 + ... + a1x + a0 is called polynomial. Here m ∈ ℕ is the degree of the polynomial if am ≠ 0, and the ai ∈ ℤ with i ∈ {0, 1, 2, ..., m} are called coefficients (of the polynomial). The complex numbers x that let the sum become zero (so-called zeros of the polynomial) are called algebraic. The corresponding sets are A in the real case and A in the complex one. In the special case am = 1 one speaks of conventionally algebraic integers. Conventionally complex numbers x that let no polynomial become zero are called transcendent.

Definition: A number a + bi with a, b ∈ ℚ is called conventionally complex-rational. A number ±b1/b2 ± ib3/b4 is called trans-complex-rational if it is not conventionally complex-rational and b2b4 ≠ 0 applies, with at least one bk that emerged from continued addition of one to |ℕ|, and conventionally integer bk else with k ∈ {1, 2, 3, 4}. A complex number that satisfies a polynomial equation p(x) = 0 of degree > |ℕ| with integer coefficients is called trans-algebraic.

In the following, the behaviour for very big n ∈ ℕ is considered.

Let be m ∈ ℕ the maximum allowable degree of the polynomial and n the maximum absolute value that the conventionally integer coefficients of the polynomials amxm + am-1xm-1 + ... + a1x + a0 with i ∈ {0, 1, 2, ..., m} are to take. This is justified since the ai are not distinguished against each other. The number of conventionally algebraic numbers corresponds to the number of zeros of the so defined normalised irreducible polynomials: The greatest common divisor gcd of their coefficients is 1 and it applies am > 0 und a0 ≠ 0.

Theorem: For the number Am of conventionally algebraic numbers (of the polynomial degree m, and thus in general) applies the asymptotical equation


where ζ is the Riemann zeta function and z(m) the (average) number of zeros of a polynomial.

Proof: To determine the number of the reducible polynomials, multiply a left polynomial factor, with coefficients whose absolute value is maximally one, successively by two and check how great the coefficients of a right polynomial may maximally become when building the product. For this O(log n) the steps are necessary and the multiplications by two are neutralised in each step by the divisions by two. The factor 1/ζ(m+1) provides for the elimination of polynomials with gcd(a0, a1, ... , am) ≠ 1. To eliminate multiples of the prime p, the multiplication of the number of polynomials by (1-p-m-1) is required. Building the product over all the primes and developing the factors into geometric series yields after multiplying the factor of 1/ζ(m+1). If exactly one coefficient is 0, ζ(m+1) would be to be replaced by ζ(m). This replacement is, however, equally covered by the correction term as the polynomials that have more than one coefficient equal to 0. In the case m = 1, the correction term O(n log n) is necessary if one calculates the number of the conventionally rational numbers via Euler's totient function as formula_002. For m > 1, the correction term O(z(m)(2n+1)mlog n) cannot be exceeded because of the divisibility circumstances, so that it is justified and the claim follows.

Remark: In the complex case, according to the fundamental theorem of algebra applies z(m) = m. In the real case, z(m) is asymptotically equal to 2/π log m + O(1) after (Mark Kac, "On the average number of real roots of a random algebraic equation. II.", Proc. London Math. Soc. 50 (1949), 390-408. MR 11:40e).

For m = 1 one obtains 12n22 + O(n log n) conventionally rational solutions. For m = 2 one obtains 4n3/ζ(3) + O(n2log n) conventionally real solutions, since a real polynomial of degree 2 has two conventionally real zeros with probability ½. For am = 1 one obtains z(m)(2n+1)m + O(z(m)(2n+1)m-1log n) conventionally algebraic integer solutions. That what was said above about z(m) here also applies.

For m = n = |ℕ|, we obtain in the real case


and in the complex case


The set of conventionally algebraic numbers contains no numbers whose absolute value is less than 1/(|ℕ|2|ℤ||ℕ|) or greater than |ℕ|2|ℤ||ℕ|. The number of the set of complex (real) numbers includes as many elements as one allows. These can be created, among other techniques, by continued exponentiation.

If one counts with m and n beyond |ℕ|, the solutions of the polynomial equations form the field of the hyper-algebraic numbers. The proofs and results are obtained as with the conventionally algebraic numbers. Analogously, there are also hyper-transcendent numbers.

The determination of the number of elements of any constructed infinite set must consider exactly its construction before it can be related to the number of conventionally natural numbers. This should be, due to its simple construction, used as basis. Without knowing the construction of a set, their number cannot be (uniquely) determined.

If there are several construction options, the most plausible should be used, i.e. it should represent infinity in the best possible way, in the sense of differentiation. Since this requires a value judgement, it has not to be uniquely determined. If one does not agree on a single construction option, despite rational reasoning, so the calculated number of elements of the set is to state with its construction.

Mathematics is not value-free. There can be always stated (rational) reasons why one procedure is to choose over the other one.

In the following, the notations from the set theory are applied. The behaviour for big m, n ∈ ℕ is considered and let be also i, j and k ∈ ℕ.

Definition: A complex number is called finitely algebraic if the set of the conventionally algebraic numbers is defined finite. It is called finitely transcendent if it is not finitely algebraic. It is called k-algebraic if it is conventionally algebraic and the k-th dual place and all succeeding of its real and imaginary part in binary representation 0. It is called k-transcendent if it is transcendent or the k-th dual place of its real or imaginary part in binary representation 1.

Thus all finitely algebraic numbers are conventionally algebraic and all transcendent numbers finitely transcendent. The inverse does not apply. It may correspond k+1 (because of the sign bit), for example, to the maximally representable binary digits of a number or its mantissa in the computer. Then the k-transcendent numbers are the ones just not (exactly) representable there.

It is known from algebra that the sum, difference, product, quotient of two finitely algebraic numbers of degree j and k are finitely algebraic of maximally degree jk, and the 1/j-th power of a finitely algebraic number of degree k is finitely algebraic of maximally degree jk.

(Finitely) transcendent numbers consist therefore especially of an (finitely) conventionally algebraic main part and a (finitely) transcendent remainder. One cannot thus simply omit the sequence values ai for big i; if one examines the (finite) transcendence of a number: They are decisive. (Finitely) transcendent numbers are all numbers that lie between the (finitely) conventionally algebraic ones and beyond them. Between two different (finitely) transcendent numbers ((finitely) conventionally algebraic numbers of degree k), lying closely enough together, is no (finitely) conventionally algebraic number (of degree < k).

Definition: A complex number x is called downward (upward) finitely bounded if there is a finitely algebraic real number c > 0 with c ≤ |x| (|x| ≤ c). A downward and upward finitely bounded complex number is called two-sided finitely bounded.

Sum proposition for finitely transcendent numbers: Every sum a|ℕ|r + v with a two-sided finitely bounded complex a, a downward finitely bounded real r and a finitely algebraic complex v is finitely transcendent.

Proof: Inserting a|ℕ|r + v into each equation of a minimal polynomial yields in each case bO(|ℕ|s) + w ≠ 0 according to the Archimedean proposition (cf. set theory) with a two-sided finitely bounded complex b, a downward finitely bounded real s and a finitely algebraic complex w.

To these finitely transcendent numbers belongs in particular, in each case, the last sequence member that comes closer to a finitely algebraic complex number than any finitely algebraic complex number, such as the number e comes so close to the number -1. The sum proposition may still be sharpened by finitely continuing to logarithmise |ℕ|r to a two-sided finitely bounded positive real base.

Bound proposition for transcendent numbers: Every complex number different from zero, whose absolute value of the real and/or imaginary part is less or equal 1/|ℕ0| or greater or equal |ℕ0|, is already transcendent.

Proof: If one sets am = 1 and ai = -|ℕ| for i < m, then the assertion follows from the geometric series, if one still builds the reciprocal. One receives the exact limit values, if one replaces |ℕ0| in both cases by |ℕ0|-µ with µ = ℕ/(|ℕ0|-µ)m. In the complex case, the statements of the form x = (1+bi)|ℕ0| for b and x ∈ ℝ yield the desired.

Coefficient proposition for (finitely) transcendent numbers: Let be n0 ∈ ℕ finite. All normalised irreducible polynomials in that is |ai| > |ℕ| (n0) for at least one ai have only (finitely) transcendent zeros.

Proof: The zeros of normalised irreducible polynomials are pairwise different and uniquely determined. Since they are not (finitely) conventionally algebraic, they must be (finitely) transcendent.

Product and sum proposition for transcendent numbers: All products that cannot be further simplified of a trans-complex-rational number g and a complex conventionally algebraic or trans-algebraic number z (here also the sum) as well as sums and products of a complex trans-algebraic and a complex conventionally algebraic (complex trans-algebraic) number with coprime polynomial degree are transcendent.

Proof: The simplification is unique, if it also includes minimising the absolute value of the coefficients of the minimal polynomial of z and of each numerator and denominator of the trans-complex-rational number g. If z is complex-rational, gz only satisfies an equation of a minimal polynomial of degree ≤ 2, with at least one coefficient with absolute value > |ℕ|. If z is irrational, gz cannot be zero of the equation of the minimal polynomial of z, since this equation is, after inserting z, not identical with the one in z, what it would have to be. Would be gz zero of another equation of a minimal polynomial, so this polynomial equation in gx could be extended by multiplication with the minimal polynomial in ḡx, with ḡ conjugated to g, to a polynomial equation in x, whose zero would be also z and where the absolute value of at least one coefficient would be > |ℕ|. This is impossible, since z can be only zero of one minimal polynomial. The involvement of a complex trans-algebraic number leads always to a equation of a minimal polynomial with at least the same polynomial degree, if the number is coprime.

Proposition: The irrational quotients of two finite sums of finite powers, multiplied by finite complex-rational numbers, with finite natural exponent of the same (finitely) transcendent number, are all finitely transcendent.

Proof: Inserting into the polynomial equations and suitable multiplying by the conjugated complex yields directly the assertion.

Together with the conventionally rational numbers, the trans-(complex-)rational numbers build numerically already complete ~ℝ (~ℂ). Therefore, (conventionally) algebraic and transcendent numbers are (numerically) hardly to distinguish and approximations are little meaningful concerning the algebraicity (of a certain degree).

Who wants to examine the transcendence of a number with Liouville's approximation theorem should ensure that the approximating rational numbers are actually still conventionally rational and not trans-rational, such as the number 10|ℕ|. Also, who considers prime numbers greater than |ℕ| in transcendence proofs should safely move in the uncountable.

Proposition: The sum of all natural powers of a complex-rational number x that is different from zero and no root of unity, (geometric series) is already transcendent.

Proof: The absolute value of the numerator or denominator of x|ℕ₀| is for |x| < 1 (|x| > 1) less than 2-|ℕ|+1 (greater than 2|ℕ|-1). The subtraction of 1 and the division by 1 - x do not change anything of the transcendence.

Proposition: Euler's number e is transcendent.

Proof: From the definition of e via the exponential series follows e = (k|ℕ|+1)/|ℕ|! with k > |ℕ|. Therewith, numerator and denominator of the stated fraction must be > |ℕ0|, since in the numerator neither |ℕ| nor a prime divisor of k can be cancelled against |ℕ|!. If one defines e as (1+1/|ℕ|)|ℕ|, so the assertion is trivial. Notice that here two different numbers are defined.

Proposition: The twin prime constant C2, as product over all (1 - 1/(p - 1)2) with prime p > 2, is transcendent.

Proof: According to the prime number theorem, the greatest and second greatest prime number - both = |ℕ| - |O(log |ℕ|)| - do not divide the denominator of C2, from which the assertion follows.

Proposition: The Landau-Ramanujan constant K, as product, divided by √2, over all (1 - 1/p2) with prime p ≡ 3 mod 4, is transcendent.

Proof: According to the (Dirichlet) prime number theorem, the greatest and second greatest prime number ≡ 3 mod 4 - both = |ℕ| - |O(log |ℕ|)| - do not divide the denominator of K, from which the assertion follows.

Proposition: The Glaisher-Kinkelin constant A = 11 22 33 ... |ℕ||ℕ| (1+1/|ℕ|)|ℕ|/4/|ℕ|(|ℕ|/2+|ℕ|/2+1/12) is transcendent.

Proof: After cancelling, the greatest prime number of ℕ remains in the denominator with exponent > 2.

Greatest-prime criterion for transcendent numbers: If a real number r has, for cancelled fractions, the representation a/(bp) ± s/t with natural a, b, s and t, abst ≠ 0 and b + t > 2, as well as the (second) greatest natural prime number p of an index set I ⊇ ℕ with p ∤ a and p ∤ t, so it is transcendent.

Proof: It applies r = (at ± bps)/(bpt) with a denominator ≥ 2p ≥ 2|ℕ| - |O(log |ℕ|)| > |ℕ0| as consequence of the prime number theorem, from which the assertion follows.

Proposition: The circular constant π is transcendent.

Proof: This follows from Wallis' product or the product representation of the gamma function for the value -½. Notice that both representations define different numbers. Alternatively, apply the greatest-prime criterion to the Leibniz series or to the arcsin(x) Taylor series for x = 1.

Proposition: The Dirichlet beta, eta and lambda function, as well as the Riemann zeta function, yield for natural arguments greater than 1 only transcendent values.

Proof: Applying the greatest-prime criterion yields directly the assertion.

Proposition: Let be s(x) for complex x the sum of xk/k over all k ∈ I, whose elements emerged by continually adding 1 to 0. If one defines Euler's constant as γ = s(1) - log |ℕ| with I = ℕ and formulates log |ℕ| as k s(½) - s(r/2k) for |ℕ| = 2k - r, k ∈ ℕ and 2k-1 > r ∈ ℕ resp. k s(½) for |ℕ| = 2k with |I| ≥ |ℕ| for a precision of O(k/2|I|), so it is trans-rational and therewith transcendent.

Proof: One obtains -log(1 - x) = s(x) + O(x|I|+1/(1 - x)) + t(x)dx for x ∈ [0, 1[ with a real function t(x) by (exact) integration (see nonstandardanalysis) from the geometric series. The assertion follows from the greatest-prime criterion for the greatest prime p and the second greatest one q by applying Fermat's little theorem (neither p and q divide simultaneously k 2k-1 - rk (+ 2k) > 0, due to the prime number theorem, nor 2 - k can be divided by p).

Remark: It remains an open question how big the set I must be in order that the formulations mentioned above are accepted with the precision stated. It is clear that one must be content with an (infinitely precise) approximation of log |ℕ|, since one cannot calculate here as precise as one wants. What is sure is that the proposition is valid for infinitely many infinity levels of I, and especially according to conventional opinion.

Approximation proposition for real conventionally algebraic numbers: Every irrational conventionally algebraic number of degree k can be approximated by a conventionally algebraic number of degree j < k with an average error that is in the real case asymptotically equal to π ζ(j+1)/(2 |ℤ|j log j).

Proof: On the conventionally real axis, the number of the conventionally algebraic numbers increases, within unchanged limits almost evenly distributed, per degree more, circa by the factor |ℤ|. The error corresponds to the distance of the conventionally algebraic numbers among each other. The non-real conventionally algebraic numbers lie less dense.

Conclusion: Two different real conventionally algebraic numbers have at least the average distance π/(|ℤ||ℕ| log |ℕ|). The precise determination of the minimal distance requires solving an infinite non-linear non-convex optimisation problem.

Proposition: The maximal distance of two real conventionally algebraic numbers ≠ 0 amounts 1/|ℕ|2 - |O(1/|ℕ|3)|.

Proof: The real conventionally algebraic numbers ≠ 0 are farthest apart around ±1. A conventionally rational number r > 1 (0 < r < 1) can be better approximated by a real conventionally algebraic x that satisfies the polynomial equation |ℕ|/r xm - |ℕ|xm-1 = 1 (xm - |ℕ|x = -|ℕ|r). For negative r applies similar. If one wants to approximate 1 by a greater real conventionally algebraic x, it must coercively satisfy a polynomial equation with am = |ℕ|-1 and a0 = -|ℕ|. If one sets still a1 = -|ℕ| and a2 = |ℕ|, follows the assertion, since the maximal distance cannot be further diminished.

The conventional differentiation and integration obliterates, by building the conventional limit, the precise distinction of transcendence and algebraicity. This is problematic, e.g. for the exact determination of zeros. Therefore, the nonstandardanalysis on this homepage goes another (more precise) way.

© 25.10.2011 by Boris Haase

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