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Preliminary remarks: Elementary concepts such as sets, elements, etc., are as defined in the relevant literature (if not defined below). Definitions are only given when they deviate from the literature or when clarification is necessary (between multiple options). 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 end of a proof is indicated by the symbol ⃞.

Definition: If every element of a set can be successively removed in equal, specific (physically) measurable time, and the total time required by the removal process is (physically) measurable, then the number of elements of this set is *finite*. If this successive removal process is not guaranteed to terminate [in (physically) measurable time], then the number of elements of the set is *infinite*. The number of elements of the set is said to be *inconcrete* if is not finite but is not guaranteed to be infinite.

Remark: Inconcrete numbers represent an intermediate stage between finite and infinite numbers. There is no justification for and therefore no sense in defining an abrupt transition between finite and infinite numbers. Sufficiently well-understood axioms exist that define the conventionally real numbers as a totally ordered field and the conventionally imaginary numbers as a field with the imaginary unit i. We can analogously extend addition, multiplication, and their inverse operations to the largest and by definition closed field extensions ℝ and ℂ := ℝ + iℝ. Other operations can of course also be extended. If the underlying set of an interval is ambiguous, it will be specified after the interval.

Definition: If the absolute value of the reciprocal of the real or imaginary parts of a non-zero complex number is infinite, this complex number is said to be *infinitesimal*. We define the *set of natural numbers* ℕ* as all numbers obtained by successively adding 1 to 0 (written ℕ when 0 is included). The set of *prime numbers* ℙ is defined by excluding all composite numbers and {0, 1} from ℕ. The set of *integers* ℤ is obtained by introducing the additive inverses of ℕ* to ℕ. The set of *rationals* ℚ is defined as the set of fractions with integer numerator and natural denominator ≠ 0.

Definition: The *number* of elements in a set M is denoted |M| and the largest *finite* natural number ć := ⌊c⌋, where c is recursively defined by c := ć + 1 - ć/c^{ć}. Let ώ := ⌊ω⌋ be the largest *non-infinite* natural and therefore *inconcrete* number. The corresponding largest non-infinite real number is recursively defined as ω := ώ + 1 - ώ/ω^{ώ} (see Transcendental Numbers). Let ^{ω}ℝ := [-ω, ω]ℝ and ^{ω}ℂ := ^{ω}ℝ + i^{ω}ℝ ⊂ ℂ. In the following, the notation ^{ω} before a set denotes the intersection with [-ω, ω]ℝ, implying that the set only contains *non-infinite* elements. The notation with ^{c} is defined analogously.

Remark: The definition (and thus limitation) of ω, clearly causes this set to lose the property of being closed. When written before a term, ^{ω} means that this term applies to non-infinite subsets instead of just to all numbers of a certain type. Although the definitions of the largest finite and non-infinite real numbers seem slightly arbitrary, we will retain these definitions due to a lack of compelling alternatives to the algebraic numbers c and ω.

Definition: The sum

where z ∈ ℂ and ά := ⌊α⌋ is called an *(α-)polynomial*, if the number of *coefficients* with e.g. a_{k} ∈ ^{c}ℤ or a_{k} ∈ ^{ω}ℤ and k ∈ ℕ_{≤ά} and a_{k} ≠ 0 is finite , otherwise *α-series*. Then deg(p) := max k for a_{k} ≠ 0 is called the *degree* of the polynomial or series p. For the *zero polynomial* p = 0, we have that deg(p) := -1. The complex numbers z ∈ ℂ for which the polynomial or series sums to zero (the *zeros* of the polynomial or series) are said to be *α-algebraic*. The sets of α-algebraic numbers are denoted ^{α}A_{ℝ} in the real case and ^{α}A_{ℂ} in the complex case. In the special case a_{deg(p)} = 1, we say that the sum is an *α-algebraic integer*. Complex numbers z ∈ ℂ that are not a zero of any α-polynomial or α-series are said to be *α-transcendental*. When α := c, this gives the conventional notion of transcendental numbers.

Definition: We can write α-algebraic numbers in the form (α, a_{m}, a_{m-1}, ..., a_{1}, a_{0}; g, h; #l, Mv; w, p)_{s}, where g = h = a_{0} = 0 represents the number 0, g ∈ ^{c}ℕ* (^{c}ℤ_{<0}) denotes the g-th largest (|g|-th smallest) zero with real part > 0 (< 0), and the other algebraic numbers have analogous notations, with g taking precedence over h. The value #l gives the quantity l ∈ ^{c}ℕ* of zeros. When at least one a_{k} is taken as a variable, Mv gives the number v ∈ ^{c}ℕ of repeated zeros. For α-minimal polynomials or series (a_{0} = 0 only for the zero polynomial), the letter m is taken as the value of the specification s, in every other case, the letter n is used. The numerical value w is given up to the precision p.

Remark: This allows the zeros of α-polynomials or α-series with integer or rational coefficients to be endowed with a strict total ordering, provided that we do not distinguish between repeated zeros. The information g, h, # l, Mv, s, w, and p can optionally be omitted (e.g. for rational numbers). The (|^{c}ℕ|+2)-tuple (0, ..., 0, a_{m}, ..., a_{0}; g, h)_{m} where the a_{k} are finite natural numbers, gives a strict lexical well-ordering of the algebraic numbers.

Definition: The sets ^{c}A_{ℝ} and ^{c}A_{ℂ} and their subsets correspond to the conventional sets A_{ℝ} and A_{ℂ} and their subsets. The set ^{c}ℝ := [-c, c]ℝ corresponds to the conventional set ℝ, the set ^{c}A_{ℂ} is defined by ^{c}A_{ℝ} + i^{c}A_{ℝ}. The set ^{c}ℚ and their subsets are defined by [-c, c]ℚ and their subsets. The set ℝ \ ^{c}ℚ denotes all *irrational* (real) numbers.

Examples: The finite numbers (c, 1, 0, 0, 0, -1)_{n} are 1, -1, i, -i. The finite golden ratio (1 + √5)/2 may be written as (c, 1, -1, -1; 1, 0; 1.618033, 10^{-6})_{m}. The number 0.1 = 0.1...1 with ώ ones after the comma is inconcrete and distinct from the finite number 1/9, since 9 × 0,1 = 0,9...9 = 1 - 10^{-ώ } ≠ 1. It is therefore ω-transcendental (cf. Transcendental Numbers) and should therefore be written as (ω, 9 × 10^{ώ }, 1 - 10^{ώ }). This implies the following result:

Theorem: For each g ∈ ℕ_{>1}, the g-adic expansion of a real number is uniquely determined, without requiring any additional conditions.⃞

Definition: A number a + bi such that a, b ∈ ℚ is said to be *complex-rational*. A number ±b_{1}/b_{2} ± ib_{3}/b_{4} is said to be *inconcrete resp. infinite complex-rational* if for k ∈ {1, 2, 3, 4}, b_{2}b_{4} ≠ 0 and |b_{k}| ∈ ℕ \ ^{c}ℕ resp. ℕ \ ^{ω}ℕ for at least one b_{k} - and the other b_{k} are integers.

Theorem: No set admits a bijection with one of its subsets.

Proof: We can prove the result by transfinite induction. We begin with a set of one element and extend to sets with multiple elements by successively adding individual elements. The same result can be obtained by removing an element from a set and attempting to find a bijection with the set thus obtained: no such bijection exists, since the missing element cannot be replaced. Transfinite induction completes the proof.⃞

Claim: The Cantor polynomial P(m, n) := ((m + n)^{2} + 3m + n)/2 is a bijective mapping from ^{ω}ℕ^{2} to ^{ω}ℕ.

Refutation: We have that P(ώ , ώ ) = 2ώ(ώ + 1) > ώ = max ^{ω}ℕ.⃞

Remark: Similarly, the Fueter-Pólya conjecture is refuted. If the set ^{ω}ℕ^{2} is replaced by {(m, n) ∈ ^{ω}ℕ^{2} : m + n ≤ k ∈ ^{ω}ℕ} for k(k + 3)/2 = ώ, the claim holds. Since any given set is isomorphic to itself, any secondary properties that can be uniquely derived are also identical. Therefore, if the secondary properties of two sets do not coincide, these two sets cannot be isomorphic. Hence, the Poincaré conjecture may be stated as:

Theorem: Every simply connected n-dimensional manifold with n ∈ ℕ is homeomorphic to a region of the n-sphere S_{r} with radius r ∈ ℝ.

Proof: Contract a net with variable mesh representing the manifold around the n-sphere.⃞

Example: For x ∈ ^{ω}ℝ_{≥1}, the mapping 1/x maps ^{ω}ℝ_{≥1} bijectively to the interval [1/ω, 1] ⊂ [0, 1], which is bounded on both sides. Therefore, we can analogously map all infinite subintervals of ℝ* bijectively to a finite interval that is bounded on both sides. Finiteness and infiniteness are therefore two ways of viewing the same isomorphic objects.

Conclusion: This in particular contradicts Dedekind-infiniteness and Cantor's first diagonal argument, since ℕ is a proper subset of ℚ. The same is true for the Banach-Tarski paradox. A translation of an infinite set always departs from the original set. This contradicts Hilbert's hotel. Regarding the continuum hypothesis, note that there are infinitely many sets whose number of elements lie between |^{c}ℕ| and |^{c}ℝ|.

Example: Whether |^{ω}ℕ| is even or odd is undecidable, since both cases are equally justified.

Remark: The symbol ∞ can be adjoined to the real and complex numbers. It may be used in calculations like a variable, and its value is formally greater than any element in ℝ. Since division by 0 is not defined in calculations due to non-uniqueness, we can simplify things e.g. by replacing ±0 by ±1/∞ wherever it makes sense to do so, depending on which direction is relevant to the present case, and continue to use ∞ in the calculations as before.

Remark: The above proceeding once again makes the calculations uniquely determined and consistent. This allows us to avoid any vague notions of limits, but we must carefully pay attention to where this replacement makes sense, and not arbitrarily switch between symbols. This will allow us to define integrals and differentials for each operation on real and complex numbers in such a way that every function is integrable and differentiable (at least directionally) wherever the function values are defined (see Nonstandard Analysis).

Definition: We define *pi* π as the area or half of the circumference of the unit circle. We define *Euler's number* e as the solution of the equation x^{iπ} = -1. We also define the *logarithm function* ln by e^{ln z} = z and the corresponding *power function* by z^{s} = e^{s ln z} for complex s and z. This allows us to give a (formal) definition of *exponentiation*. In calculations, we will typically need to resort to approximations.

Remark: By the binomial theorem, the alternative definition of e := (1 + 1/ć)^{ć} is O(1/c) smaller than the above, which can be seen by considering the exponential series with as many terms as possible (and differentiating exactly). This deviation can have negative consequences when attempting to calculate e as precisely as possible.

Lemma: There are infinitely many numbers in ^{c}ℝ for which the Archimedean axiom does not hold.

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

Archimedes' theorem: There exists n ∈ ^{c}ℕ such that b n > a if and only if a/b < ć whenever a > b for a, b ∈ ℝ_{>0}.

Proof: If a/b ≥ ć, then a/b ≥ n, for all n ∈ ^{c}ℕ.⃞

Remark: Let m ∈ ^{c}ℕ be the maximum polynomial degree and n ∈ ^{c}ℕ the maximum absolute value that the integer coefficients a_{k} of the polynomials a_{m}x^{m} + a_{m-1}x^{m-1} + ... + a_{1}x + a_{0} with k ∈ ^{c}ℕ_{≤m} can take. This makes sense due to the symmetry of the a_{k}. The number of algebraic numbers is the number of zeros of the normalised irreducible polynomials specified by the conditions: greatest common divisor (gcd) of the coefficients is equal to 1, a_{m} > 0 and a_{0} ≠ 0.

Counting theorem of algebraic numbers: The number A_{m} of algebraic numbers (of polynomial or series degree m and thus in general) asymptotically satisfies the equation

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

Proof: The case m = 1 is proven in [455] and the error term O(n ln n) is required when estimating the number of rational numbers with the Euler φ-function by . Due to divisibility conditions, the error term does not exceed O(z(m)(2n+1)^{m}ln n) for m > 1, and the leading term does not change. The factor of 1/ζ(m + 1) eliminates all polynomials such that gcd(a_{0}, a_{1}, ... , a_{m}) ≠ 1. To remove repeat prime numbers p, we must multiply the number of polynomials or series by (1 - p^{-m-1}). Taking the product over all prime numbers and developing the factors into geometric series gives the 1/ζ(m + 1) after multiplying out. If precisely one coefficient is 0, ζ(m + 1) can be replaced by ζ(m). This is absorbed by the error term, as well as the cases corresponding to polynomials or series where more than one coefficient is 0. The result follows.⃞

Remark: In the complex case, by the fundamental theorem of algebra (see Nonstandard Analysis), z(m) = m. In the real case, z(m) is asymptotically equal to 2/π log m + O(1) according to (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) rational solutions. For m = 2, we obtain 4.5n^{3}/ζ(3) + O(n^{2}ln n) real solutions, since a real polynomial of degree 2 has two real zeros with a probability 9/16 by the quadratic formula. For a_{m} = 1, we obtain z(m)(2n + 1)^{m} + O(z(m)(2n + 1)^{m-1}ln n) algebraic integer solutions. The above remarks about z(m) are also valid here.

Examples: For m = n = ć, in the real case, omitting the ^{c} before each set, we obtain

and in the complex case

Remark: This means that the set of algebraic numbers does not contain any numbers with absolute values smaller than 1/(|ℤ|^{|ℕ*|}|ℕ|^{2}) or larger than |ℤ|^{|ℕ*|}|ℕ|^{2}.

Remark: When counting the number of elements of a finite (infinite) set, we must pay careful attention to its construction before we compare it to ^{c}ℕ or ^{ω}ℕ. These latter sets may be taken as a basis thanks to their simple constructions. If we do not know the construction of a set, it cannot be (uniquely) counted. If there are multiple possible constructions, we should choose the most plausible, i.e. the one that best reflects the finiteness (infiniteness) of the set for the purpose of differentiating between these two cases. Since this involves a value judgment, it may not be uniquely determined or determinable. If one single construction cannot be chosen after rational discussion, the construction should be indicated whenever the number of elements of the set is given.

Definition: We replace ℍ either by ℝ or ℂ. Two different points p and q in a subset M ⊆ ℍ^{n} where n ∈ ℕ* are said to be *neighbours* if ||p - q|| ≤ max (||p - r||, ||q - r||) holds for all points r ∈ M, where ||·|| denotes the Euclidean norm. The subsets of ℍ^{n} such that all neighbouring elements have the *symbolic minimum distance d0*, the smallest positive number in ℝ, are said to be *gapless*.

Definition: A non-empty subset M ⊆ ℝ is said to be *h-homogeneous* if the minimum distance between any two of its elements is h ∈ ℝ_{>0}. We denote this by *h-M*. An n-dimensional subset M ⊆ ℝ^{n} with n ∈ ℕ* is said to be h-homogeneous if it is h-homogeneous in each dimension. We define h-homogeneity analogously for subsets of ℂ^{n}. The subset M ⊆ ℍ^{n} is said to be *dense* in ℍ^{n}, if there is a point y ∈ M for every x ∈ ℍ^{n} with ||x - y|| ≤ d0.

Remark: To h-homogenise a set, we move h away from its minimal elements in each dimension and round elements in between up or down to the nearest h-homogeneous elements. Moreover ^{c}A_{ℚ} ⊂ ^{c}ℚ is true and also the inhomogeneity of ^{c}A_{ℂ} ⊂ ^{c}ℂ.

Theorem: ℚ is dense in ℝ.

Proof: There exists always the irrational number a + (b - a)/√2 between a, b ∈ ℚ with a < b. There exists always the rational number k2^{n} with d - c > 2^{n} and suitable k, n ∈ ℤ between c, d ∈ ℝ \ ℚ with c < d. We have therefore |x - y| = d0 with an irrational x and a rational y such that x and y are neighbours. We stop dividing when rational and irrational numbers alternate on the real line.⃞

Remark: We have therefore shown that ℚ numerically is almost ℝ. When we are content with the precision 2^{-n} for arbitrary large n ∈ ℕ, we obtain the h-homogeneity of ℝ with h := 2^{-n-1}. Equating |^{c}ℝ| with 2^{n+1} for an n ∈ ℕ means that all finite real numbers (approximately) can be developed into dual numbers. Therefore, ℕ, ℤ, ℚ, ℝ and ℂ are h-homogeneous. This is exactly true for ℚ if we convert the numbers of ℚ to a common denominator and assume that all gaps in the numerators are closed. Contrary to the conventional opinion, the set ^{c}A_{ℝ} is neither gapless nor dense in ^{c}ℝ.

Definition: A sequence of neighbouring h-cubes that enclose but do not contain at least one h-square is called a *closed chain* of h-cubes. An h-homogeneous set A ⊆ ^{(ω)}ℍ^{n} is said to be *simply connected*, if it does not contain any closed chain of h-cubes. It is said to be *connected*, if every h-cube is linked to every other h-cube. A simply connected h-homogeneous set A ⊆ ^{(ω)}ℍ^{n} is said to be *h-convex* if the points on the line connecting any two vertices of two separate h-cubes are also vertices of the corresponding h-cubes in A. It is said to be *h-star domain* if the points on the line between the (central) vertex of some h-cube of A and any other h-cube of A are also vertices of the corresponding h-cubes. Each individual h-cube is both h-convex and an h-star domain.

Definition: A (simply connected) h-homogeneous subset of ℝ^{m}, where m ∈ ℕ and h ∈ ℝ_{>0}, is *n-dimensional* with m ≥ n ∈ ℕ if and only if it contains at least one n-cube with edge length h and maximum n. Singletons are 0-dimensional, larger sets at least one-dimensional. The definition for ℂ^{m} is analogous.

Definition: The set ∁A := X \ A, for A ⊆ X, where X as an arbitrary set, is called *complement* of A in X. If X is clear from the context, it can be omitted and ∁A can be called the *exterior* of A. The set ∂A consists of all points of A that have a neighbour in ∁A and is called the *(inner) boundary* of A. If a set does not have a complement, the boundary contains the outermost elements, if they exist. When ∂ is successively applied beyond that, the argument is assumed to be without complement.

Definition: The set ℰA := ∂∁A is called the *outer boundary* of A. The set A° := A \ ∂A is called the *interior* of A. The set B_{(≤)r}(a) := {z ∈ H := ^{(ω)}ℍ^{n} : ||z - a|| ≤ r} is called a *ball* with *radius* r ∈ ^{(ω)}ℝ around its *centre* a ∈ H and its boundary is called a *sphere*. When a = 0 and r = 1, we obtain the *unit ball*, with the special case of the *unit disc* đ for n = 2. The set B_{<r}(a) := {z ∈ H : ||z - a|| < r} = B_{r}(a)° is accordingly called the *inner ball*.

Remark: The set ℝ of all real numbers has both a fixed minimum element and a fixed maximum element, since we view ℝ holistically and completely. Otherwise, we maybe would have to adapt our theory again and again to the circumstances. Many of the conclusions derived here can be extended to sets with other upper and lower bounds. We will not list the results for which this is possible, since the associated arguments are easy.

Further thoughts leading into physics: The definition of homogeneity of ^{c}ℝ^{n} suggests that individual directions may be distinguished from each other, making it possible to move faster in one direction than in others. However, if we define the required time in one direction to be proportional to the (Euclidean) distance from the starting point, the problem resolves itself.

© 2002-2016 by Boris Haase

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