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Set Theory

Set Theory

Preliminary remark: We presuppose basic terms (in the following not defined) such as set, element, etc., as described in the relevant literature, as known and therefore only provide different definitions or such ones to be clarified (from several possible ones) in the following. Parts of a sentence in brackets may, different from the conventional use of the more detailed explanation, either be or not: both meanings are valid. Proofs are completed with ⃞.

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. It is inconcrete, if it is not finite and also not with security infinite.

Remark: Inconcrete numbers form an intermediate stage between the finite and infinite numbers. It makes no sense to define an abrupt transition from finite to infinite numbers, since this can only be reasoned with difficulties. Sufficiently known axioms define the conventionally real numbers as totally ordered field and the conventionally complex numbers with additional imaginary unit i as field. We can continue addition, multiplication and their inverting analogously in the most comprehensive superfields ℝ and ℂ := ℝ + iℝ, closed per definition, of both. An extension to further operations is possible of course. If the support of an interval is unclear, it follows the interval.

A complex number ≠ 0 with infinite absolute value of the reciprocal of its real or imaginary part is called infinitesimal. We define all natural numbers ℕ* as all sums emerging from successive additions of 1 to 0 (including 0 as ℕ), all prime numbers ℙ by excluding composite numbers and {0, 1} from ℕ, all integer numbers ℤ by adding the additive inverses of ℕ* to ℕ, all rational numbers ℚ by fractions with integer numerator and natural denominator ≠ 0. The number of all elements of a set M is denoted with |M|.

Definition: The sum

formula_008

with z ∈ ℂ is called (α-)polynomial, falls ⌊α⌋ ∈ ℕ (the degree of the polynomial) is finite with a⌊α⌋ ≠ 0 or the sum is 0 for all z. The ak ∈ [-⌊α⌋, ⌊α⌋]ℤ with k ∈ ℕ≤ ⌊α⌋are called coefficients (of the polynomial). The complex numbers z ∈ ℂ that let the sum become zero (so-called zeros of the polynomial) are called α-algebraic. The corresponding sets are called αA in the real case and αA in the complex one. The greatest finite natural number is denoted by ⌊κ⌋. The corresponding greatest finite real κ-algebraic number is recursively defined as κ := ⌈κ⌉ - ⌊κ⌋/κ⌊κ⌋.

Definition: Let be ⌊ω⌋ the greatest non-infinite natural and therewith inconcrete number. The corresponding greatest finite real ω-algebraic number is like κ recursively defined as ω := ⌈ω⌉ - ⌊ω⌋/ω⌊ω⌋ (see transcendental numbers). If ⌊α⌋ is inconcrete, then the sum is called α-series. We speak of algebraic integers in the special case a⌊α⌋ = 1. Complex numbers z ∈ ℂ that let no α-polynomial become zero are called α-transcendental. For α := κ, we have conventional transcendence. Let be ω ℝ := [-ω, ω]ℝ and ω ℂ := ω ℝ + iω ℝ ⊂ ℂ. A preceding ω previous to a set means in the following apart from that the intersection with [-ω, ω]ℝ and that this set contains only non-infinite elements. Analogously the meaning of κ is defined.

Remark: By definition (and hence limitation) via ω, this set obviously loses its closure. Behind a term, ω means that this term applies, instead for all numbers of a type, also for non-infinite subsets. Although something arbitrary clings to the definitions of a greatest finite and non-infinite real number, we stick to both unless there are convincing alternatives.

Definition: We can denote α-algebraic numbers by (α, am, am-1, ..., a1, a0; g, h; #l, Mv; w, p)s, whereby we represent the number 0 with g = h = a0 = 0, with g ∈ κℕ* (κ<0) a zero with the gth-greatest (|g|th-smallest absolute value of the) real part > 0 (< 0), with g = 0, h ∈ κℕ* (κ<0) a non-real zero with the hth-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 ∈ κℕ* of zeros if at least for one ak a variable is used, Mv denotes the number v ∈ κℕ of multiple zeros. With a minimal polynomial (with a0 = 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 (|κℕ|+2)-tuple (0, ..., 0, am, ..., a0; g, h)m with natural ak form a lexical strict well-ordering for the algebraic numbers.

Definition: The sets κℕ, κℙ, κℤ, κℚ, κℝ and κℂ correspond to the conventional sets ℕ, ℙ, ℤ, ℚ, ℝ and ℂ and are formed, except ℕ, ℙ, ℤ, ℚ, ℝ und ℂ, from the latter by intersection with [-κ, κ]ℝ, and are preceded by κ. The set ℝ \ κℚ denotes all irrational (real) numbers.

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

Theorem: For all g ∈ ℕ>1, the g-adic expansion of a real number is uniquely determined, without additional conditions.⃞

Definition: A number a + bi with a, b ∈ ℚ is called complex-rational. A number ±b1/b2 ± ib3/b4 is called inconcrete resp. infinite complex-rational if for k ∈ {1, 2, 3, 4} with b2b4 ≠ 0 for at least one bk |bk| ∈ ℕ \ κℕ resp. ℕ \ ωℕ applies - with integer bk else.

Remark: |κℝ| = 2⌊κ⌋ states that all finite 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 transfinite 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. transfinite induction proves the rest.⃞

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. Therewith, the Poincaré conjecture must read:

Theorem: Every simply connected n-dimensional manifold with n ∈ ℕ is homeomorphic to a domain on the n-sphere Sr with radius r ∈ ℝ.

Proof: We contract a net representing the manifold with extensible meshes on the n-sphere.⃞

Example: The map 1/x maps ω ≥1 for x ∈ ω≥1 bijectively onto the both-sided finitely limited interval [1/ω, 1] ⊂ [0, 1]. Thus we can analogously map all infinite subintervals of ℝ* bijectively onto a both-sided finitely limited interval. Finiteness and infinity are thus two views of that what is isomorphic to each other.

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 |κℕ| and that of |κℝ|.

Example: It is undecidable, whether |ωℕ| is even or not, since both is equitable.

Remark: The symbol ∞ can be adjoined to the complex and real numbers, with which can be calculated as with a variable and whose value is to be symbolically greater than each element of ℝ. 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 = -1. Then, let a logarithm function ln be defined by eln z = z and the corresponding power function by zs = es 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/⌊κ⌋)⌊κ⌋ is O(1/κ) 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 κℝ.

Proof: Let a ∈ κ>1 and b = 1/κ. Then it applies b n ≤ 1 < a for all n ∈ κℕ.⃞

Archimedean theorem: There is n ∈ κℕ with b n > a, iff for a > b with a, b ∈ ℝ>0 a/b < ⌊κ⌋ applies.

Proof: If a/b ≥ ⌊κ⌋ applies, then also a/b ≥ n is valid for all n ∈ κℕ.⃞

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

Remark: Let be m ∈ κℕ the maximum allowable degree of the polynomial and n the maximum absolute value that the integer coefficients ak of the polynomials amxm + am-1xm-1 + ... + a1x + a0 with k ∈ κ≤m are to take. This is justified since the ak 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 am > 0 and a0 ≠ 0.

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

formula_005

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 formula_002. Because of the divisibility circumstances, for m > 1, the correction term O(z(m)(2n+1)mln 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(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 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 12n22 + O(n ln n) conventionally rational solutions. For m = 2 we obtain 4.5n3/ζ(3) + O(n2ln n) real solutions, since a real polynomial of degree 2 has two real zeros, according to the quadratic formula with probability 9/16. For am = 1 we obtain z(m)(2n + 1)m + O(z(m)(2n + 1)m-1ln n) algebraic integer solutions. That what was said above about z(m) here also applies.

Examples: For m = n = ⌊κ⌋, we obtain in the real case, while dropping κ before the sets,

formula_006

and in the complex case

formula_007

Remark: The set of algebraic numbers contains no numbers whose absolute value is less than 1/(|ℤ||ℕ*||ℕ|2) or greater than |ℤ||ℕ*||ℕ|2.

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 κℕ or ω 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: A one-dimensional set M ⊆ (~)ℝ is called h-homogeneous if every minimal distance of its different elements is h ∈ ~ℝ, and is denoted h-M. An n-dimensional set M ⊆ ℝn with n ∈ ℕ* is called h-homogeneous, if it is this in each dimension. In subsets of ℂn = (ℝ + iℝ)n, the definitions are analogous.

Remark: If we h-homogenise a set, we mark out h proceeding from its minimal elements in each dimension and round elements lying between these numbers onto the h-homogeneous elements.

Definition: Two points p and q of a (h-homogeneous) subset M ⊆ ℝ are called neighbour if there is no further point r ∈ M between them. Two points p and q of a (h-homogeneous) subset M ⊆ ℝn with n ∈ ℕ* are called neighbour if they are neighbour in one dimension. In subsets of ℂn = (ℝ + iℝ)n, the definitions are analogous.

Definition: Two neighbour points in ℝ have the symbolic minimal distance h = d0, which corresponds to the smallest positive number in ℝ. The (h-homogeneous) subset of ℂn, for which all neighbour elements have the minimal distance d0, are called gapless.

Remark: The sets κℚ, κA and κA are not gapless in contrast to the conventional view.

Homogeneity lemma: The set ℝ of all real numbers is not h-homogeneous for a fixed h ∈ ℝ. Nevertheless, it is (d0-) homogeneous.

Proof: If it were for two numbers a, a + h ∈ ℝ, then build a + h/2 ∈ ℝ. Contradiction! If we continue this division process in the whole ℝ, we get the homogeneity of ℝ.

Definition: A (simply connected, see nonstandardanalysis) h-homogeneous subset ℝm with m ∈ ℕ* and h ∈ ℝ is n-dimensional with m ≥ n ∈ ℕ* iff it contains at least one n-cube with edge length h and this n is maximal.

Remark: The set ℝ of all real numbers has neither a fixed minimal nor a fixed maximal element, since we can build always a smaller or greater one. The latter has priority over a total view of ℝ, since in mathematics as humanities philosophically potentiality is above act. Many of statements mentioned here can be transferred onto sets with other upper and lower limits. Where this can take place is not revealed here in detail, since the considerations for this are so easy.

Prospect into physics: The definition of the homogeneity of κn suggests that individual directions are distinguished before each other, so that we can move in one direction faster than in another. If, however, we define that the time required in one direction is proportional to the (Euclidean) distance from the starting point, this problem is resolved itself.

© 2002-2016 by Boris Haase


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