Preliminary remark: Sufficiently known axioms define the conventionally real numbers ℝ (as totally ordered field) and the conventionally complex numbers ℂ with additional imaginary unit i. We can continue addition, multiplication and their inverting analogously in supersets of both. Closure, however, does not apply (see below). An extension to further operations is possible. We define the natural numbers ℕ* as all sums emerging from an unspecified minimally infinite number of successive additions of 1 to 0 (including 0 as ℕ), prime numbers ℙ by excluding composite numbers and {0, 1} from ℕ, integer numbers ℤ by adding the additive inverses of ℕ* to ℕ, rational numbers ℚ by fractions with integer numerator and natural denominator ≠ 0. We note sets defined in this way like the conventional ones. We mark statements that apply also to the latter by ^{+}.

Remark: To demand minimal infinity is problematic, but necessary to limit closer. The number of all elements of a set M is denoted with |M|. The absolute value of the real and imaginary part of all complex numbers ≠ 0 be ≤ ω resp. ≥ 1/ω with ω := |ℕ| - |ℕ*|/ω ^{|ℕ*|} = max ℝ (see transcendental numbers). Sets of all number of a type are preceded by ^{~}, (trans-) natural and (trans-) rational numbers whose modulus is greater than |ℕ*| and (trans-) real and (trans-) complex (algebraic, see next paragraph) numbers, whose modulus of the real or Imaginary part is greater than ω, by trans- and their corresponding sets by ^{T}. Since ℝ is not closed in this way, the completeness axiom must be dropped (see below). It is thus ℝ := [-ω, ω]^{~}ℝ and ℂ := ℝ + iℝ.

Definition^{+}: The sum p(x) = a_{m}x^{m} + a_{m-1}x^{m-1} + ... + a_{1}x + a_{0} is called *polynomial*. Here m ∈ ℕ* is the *degree* of the polynomial if a_{m} ≠ 0, and the a_{k} ∈ ℤ with k ∈ ℕ_{≤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_{ℝ} in the real case and A_{ℂ} in the complex one. We speak of *algebraic integers* in the special case a_{m} = 1. Conventionally complex numbers x that let no polynomial become zero are called *transcendental*.

Definition^{+}: We can denote algebraic numbers by (a_{m}, a_{m-1}, ..., a_{1}, a_{0}; g, h; #l, Mv; w, p)_{s}, whereby we represent the number 0 with g = h = a_{0} = 0, with g ∈ ℕ* (ℤ_{<0}) a zero with the g^{th}-greatest (|g|^{th}-smallest absolute value of the) real part > 0 (< 0), with g = 0, h ∈ ℕ* (ℤ_{<0}) a non-real zero with the h^{th}-greatest (|h|^{th}-smallest. absolute value of the) imaginary part > 0 (< 0), and the remaining algebraic numbers are so correspondingly, where g has priority over h. The notation #l specifies the number l ∈ ℕ* of zeros if at least for one a_{k} a variable is used, Mv denotes the number v ∈ ℕ of multiple zeros. With a minimal polynomial (with a_{0} = 0 only for the polynomial x = 0) for the specification s the letter m is denoted, otherwise the letter n. We represent the numeric value by w with precision p.

Remark^{+}: In this way the zeros of a polynomial with integer or rational coefficients can be strictly totally ordered, if multiple zeros are not distinguished. The particulars g, h, # l, Mv, s, w and p can be omitted as the case may be (e.g. for rational numbers). The (|ℕ|+2)-tuple (0, ..., 0, a_{m}, ..., a_{0}; g, h) _{m} with natural a_{k} form a lexical strict well-ordering for the algebraic numbers.

Definition: If all elements of a set can be successively removed in always the same determined (physically) measurable time and if the total time of removing is (physically) measurable, then the number of elements of this set is *finite*. If this successive removing cannot come to an end [in (physically) measurable time], the number of the elements of this set is *infinite*.

Definition: Algebraic numbers that are denoted by finitely many concrete figures, according to the previous definition, are called *concretely given*, the undetermined many ones that can be denoted in this way *concretely specifiable*. Both together form the subset of *concrete* numbers, which is signed as well as the subsets of algebraic numbers with the prefix ^{c}. With κ := max ^{c}A_{ℝ}, the real and imaginary parts of all *concrete complex* numbers have a modulus in the interval [0, κ]. The sets ^{c}ℕ, ^{c}ℙ, ^{c}ℤ, ^{c}ℚ, ^{c}ℝ and ^{c}ℂ correspond to the conventional sets ℕ, ℙ, ℤ, ℚ, ℝ and ℂ. The remaining algebraic numbers are called *inconcrete*, and the prefix symbol is ^{u}. (Trans-) Real numbers that are not conventionally rational are called *irrational*.

Definition: (Trans-) complex numbers whose modulus of the real and imaginary part is ≥ |ℕ*| are called *infinite*: the preceding symbol is ^{∞}. A complex number ≠ 0 with infinite absolute value of the reciprocal of its real or imaginary part is called *infinitesimal*.

Remark: Infinitesimal numbers and those whose modulus is maximally a finite natural number are finitely bounded. Inconcrete numbers form an intermediate stage between the (in each case finite) concrete and the infinite numbers. If the carrier of an interval is unclear, it follows the interval.

Examples: The concretely specifiable numbers (1, 0, 0, 0, -1)_{n} are given as 1, -1, i, -i. The concretely given golden number (1 + √5)/2 can be denoted as (1, -1, -1; 1, 0; 1.618033, 10^{-6})_{m}. The number 0.1 = 0.11...11 with |ℕ*|ones after the comma is inconcrete and different from the concretely given number1/9, since 9 × 0.1 = 0.99...99 = 1 - 10^{-|ℕ*|} ≠ 1. Hence it is transcendental (cf. transcendental numbers) and would have to be denoted for instance as (9 × 10^{|ℕ*|}, 1 - 10^{|ℕ*|}). Hence, we have the

Theorem^{+}: For all g ∈ ℕ_{>1}, the g-adic expansion of a (trans-) real number is uniquely determined, without additional conditions.

Definition^{+}: A number a + bi with a, b ∈ ℚ is called *complex-rational*. A number ±b_{1}/b_{2} ± ib_{3}/b_{4} is called *trans-complex-rational* if for k ∈ {1, 2, 3, 4} with b_{2}b_{4} ≠ 0 for at least one b_{k} |b_{k}| ∈ ^{~}ℕ_{≥|ℕ|} applies - with conventionally integer b_{k} else. The (trans-) complex numbers that satisfy *polynomial equations* p(x) = 0 of degree or with at least one integer coefficient of absolute value ≥ |ℕ| form with the algebraic ones the field of *hyper-algebraic* numbers with infinitely many subfields. The proofs and results are obtained as for the algebraic numbers. Analogously, there are also *hyper-transcendental* numbers.

Remark^{+}: |ℝ| = 2^{|ℕ|} states that all real numbers can be developed into dual numbers. Against this speaks, on the one hand, , that we can represent many rational numbers only approximately as dual numbers and much more that 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.

Theorem^{+}: For no set, there is a bijection to its proper subset.

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

Remark^{+}: Since a set is isomorphic to itself, also its secondary properties are, as long we can deduce these uniquely. If the latter thus do not match, two sets cannot be isomorphic.

Conclusion^{+}: Thus especially Dedekind-infinity and Cantor's diagonal argument are discounted, since ℕ is a proper subset of ℚ. The same is valid for the Banach-Tarski paradox. 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 natural numbers is not closed concerning addition.

Proof: If we form by bijection to ℕ the set 2ℕ, so this contains elements that cannot be contained in ℕ, since the odd numbers are missing in 2ℕ. On the other hand, for all k ∈ ℕ also k + k must be included in ℕ. This contradiction implies the assertion.

Conclusion^{+}: The rational, algebraic, real and complex numbers as well as their supersets are also not closed concerning addition, because they are based on the addition of conventionally natural numbers, resp. then this cannot be anymore be appropriately defined for some elements, because it cannot go beyond |^{~}ℕ|. The lack of completion is also transferable to multiplication and inversion, since they build on the addition. Sum, difference, product and quotient of two concretely given numbers are always concretely specifiable. All this results especially from the following

Minimax theorem^{+}: 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 and such with inconcretely naturally many elements are isomorphic to sets consisting only of consecutive ordinal numbers - starting with 0. Consider now the set [0, 2^{|ℕ|}]^{~}ℕ. This is isomorphic to the homogeneous set ([0, 2^{|ℕ|}]^{~}ℕ)/2^{|ℕ|}, which 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|} for k ∈ [0, 2^{|ℕ|}]^{~}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 2^{k} ↦ 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, 2^{|ℕ|}]^{~}ℕ)/2^{|ℕ|} with 1/2^{|ℕ|}-steps, and for all bigger ordinal number sets by successively still increasing the exponent |ℕ| by 1 and proceeding analogously.

Self-evident indefiniteness lemma: About the minimal and maximal elements of an inconcrete or infinite set, only a more precise 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.

Definition: The smallest inconcrete element greater than κ is denoted as κ_{+}, the smallest infinite one greater than ω as ω_{+} and the absolutely greatest one max ^{~}ℝ as ω_{∞}.

Remark: The definition is indeed problematic, but useful.

Example^{+}: The sets ℕ and ℕ ∪ {|ℕ|} can be called sets of natural numbers, having equal rights. Therefore it is undecidable, whether |ℕ| is even or not.

Remark^{+}: The virtual (see below) symbol ∞ can be adjoined to the (trans-) complex and (trans-) real numbers, with which can be calculated as with a variable and whose value is to be greater than ω_{∞}. Since division by 0 is not defined, due to the lack of uniqueness in calculations, we can manage this with replacing, wherever it is convenient, for example, ±0 by ±1/∞, depending on which direction we are interested in, and calculating with ∞ as described above. Then calculating is again unique and consistent. Considering vague limits can be avoided, but every replacement should be carefully considered where it makes sense, and it should not be arbitrarily switched between the symbols. In this way, integral and differential for each operation on the complex and real numbers can be defined such that each function is everywhere integrable and differentiable (at least as a directional derivative), where the values of the function are determined (see nonstandardanalysis).

Definition: Let the circular constant π be defined as area or the half circumference of the unit circle. Let *Euler's number* e be defined as solution of the equation x^{iπ} = -1. Then, let a *logarithm function* ln be defined by e^{ln z} = z and the corresponding *power function* by z^{s} = e^{s ln z} with complex s and z. In this way, the *exponentiation* can be defined (symbolically). Computationally, we will usually have to be content with approximations.

Remark^{+}: The possible definition of e := (1 + 1/|ℕ*|)|ℕ*| is O(1/|ℕ|) smaller than the preceding one, which is justified by the exponential series with as many terms as possible (per exact differentiation). This deviation can have a negative effect, if we want to calculate as precise as possible.

Lemma: The Archimedean property does not apply for infinitely many real numbers from ^{(c)}ℝ.

Proof: Let a ∈ ^{(c)}ℝ_{>1} and b = 1/|^{(c)}ℕ*|. Then applies b n ≤ 1 < a for all n ∈ ^{(c)}ℕ.

Archimedean theorem: There is n ∈ ^{(c)}ℕ with b n > a, iff for a > b with a, b ∈ ℝ_{>0} a/b < |^{(c)}ℕ*| applies.

Proof: If a/b ≥ |^{(c)}ℕ| applies, then also a/b ≥ n is valid for all n ∈ ^{(c)}ℕ.

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

Remark^{+}: Let be m ∈ ℕ the maximum allowable degree of the polynomial and n the maximum absolute value that the conventionally integer coefficients a_{k} of the polynomials a_{m}x^{m} + a_{m-1}x^{m-1} + ... + a_{1}x + a_{0} with k ∈_{≤m} are to take. This is justified since the a_{k} are not distinguished against each other. The number of 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 a_{m} > 0 and a_{0} ≠ 0.

Number theorem of algebraic numbers^{+}: For the number A_{m} of 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: The case m = 1 applies according to [455] and the correction term O(n ln n) is necessary if we calculate the number of the conventionally rational numbers via Euler's totient function as . Because of the divisibility circumstances, for m > 1, the correction term O(z(m)(2n+1)^{m}ln n) cannot be exceeded, and the main term does not alter. The factor 1/ζ(m+1) provides for the elimination of polynomials with gcd(a_{0}, a_{1}, ... , a_{m}) ≠ 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 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, so that the claim follows justly.

Remark^{+}: In the complex case, according to the fundamental theorem of algebra (see nonstandardanalysis) 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).

Examples^{+}: For m = 1 we obtain 12n^{2}/π^{2} + O(n ln n) conventionally rational solutions. For m = 2 we obtain 4n^{3}/ζ(3) + O(n^{2}ln n) conventionally real solutions, since a real polynomial of degree 2 has two conventionally real zeros with probability ½. For a_{m} = 1 we obtain z(m)(2n+1)^{m} + O(z(m)(2n+1)^{m-1}ln n) algebraic integer solutions. That what was said above about z(m) here also applies.

Examples^{+}: For m = n = |ℕ|, we obtain in the real case

and in the complex case

Remark^{+}: 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 continued exponentiation allows.

Remark^{+}: The determination of the number of elements of any (in-) finite set must consider exactly its construction before we can relate it to the set ℕ. This should be, due to its simple construction, used as basis. Without knowing the construction of a set, its number cannot be (uniquely) determined. If there are several construction options, we should use the most plausible one, i.e. it should represent (in-) finiteness in the best possible way, in the sense of differentiation. Since this requires a value judgement, it has not to be uniquely determined. If there is no agreement on a single construction option, despite rational reasoning, so the calculated number of elements of the set is to state with its construction.

Definition: We define sets like ^{(~)}ℚ, A_{ℝ}, A_{ℂ}, ^{(~)}ℝ and ^{(~)}ℂ in the nonstandardanalysis homogenous by letting determine the number of the elements of the sets equally the homogenous distances of the elements, where alone ^{(~)}ℝ and the real resp. imaginary part of ^{(~)}ℂ are arbitrarily dense. For (hyper-) (algebraic) subsets of ^{~}ℝ, we can determine this distance via the minimal distance h ∈ ^{~}ℝ of the (hyper-) algebraic numbers. Here, in the h-homogeneous sets, only the transcendental numbers are that emerge by marking out h from the two selected and maybe minimal adjacent (hyper-) algebraic numbers to be specified. Herewith we approve the impreciseness, when (hyper-) algebraic numbers are between these points marked out. The deviation from the exact values maybe still further specified. Since it is infinitesimal, this is, however, less important. We distinguish only between this, each time, next approximations and the proper elements, if this is explicitly mentioned. This can lead to the fact that the inverses of a set are not in the set anymore. Then |^{(~)}ℕ*| = |^{(~)}ℝ| = |^{(~)}ℤ_{>0}| applies. The set ^{~}ℝ is *(trans-) really countable*, as well as its subsets. We consider equally for mappings from one domain into its codomain only the images that lie in the codomain as described by specifying suitable elements as images for this purpose.

Definition: Since it can be necessary to use smaller or greater numbers as well as their multiples as auxiliary numbers than there are in ^{~}ℝ, they form altogether the set of *virtual* numbers. A (simply connected) h-homogeneous point set with h ∈ ^{~}ℝ (see nonstandardanalysis) from ^{~}ℝ^{|˜ℕ*|} is exactly *n-dimensional* with n ∈ ^{~}ℕ*, iff it contains at least an n-cube with edge length h and this n is maximal.

Examples: |^{~}ℤ|, |^{~}ℕ|/d0 and ω_{∞} + 1 and their are virtual numbers.

Remark: The maximal number of real dimensions is |^{~}ℕ*|. |^{~}ℂ|^{|˜ℕ*|} has thus |^{~}ℤ*| virtual many real dimensions. The set ^{~}ℂ is only (trans-) virtual countable. Obviously we have ^{~}ℚ ⊂ ^{~}ℝ, aber ^{~}ℕ ⊄ ^{~}ℚ. Since we can determine the number of elements of a set now exactly, many statements of linear algebra that apply for finite-dimensional vector spaces are also valid for infinite-dimensional ones.

© 17.11.2015 by Boris Haase

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