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Set Theory

Set Theory

Definition: Let two entities be identical if we do not want to distinguish them. The entirety of non-identical entities is called set of elements. 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, otherwise inconcrete or infinite, dependent on their indeterminate resp. not fulfilled measurability.

Remark: Inconcrete numbers represent an intermediate stage between finite and infinite numbers. An abrupt transition between finite and infinite numbers is only hard to justify, however. 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 \(\mathbb{R}\) and \(\mathbb{C} := \mathbb{R} + i\mathbb{R}\). Other operations can also be extended.

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 \(\mathbb{N}^{*}\) as all numbers obtained by successively adding 1 to 0 (written \(\mathbb{N}\) when 0 is included). The set of prime numbers \(\mathbb{P}\) is defined by excluding all composite numbers and {0, 1} from \(\mathbb{N}\). The set of integers \(\mathbb{Z}\) is obtained by introducing the additive inverses of \(\mathbb{N}^{*}\) to \(\mathbb{N}\). The set of rationals \(\mathbb{Q}\) is defined as the set of fractions with integer numerator and natural denominator \(\ne 0\). The set of complex-rational numbers is \(\mathbb{Q} + i\mathbb{Q}\).

Theorem: For every \(b \in \mathbb{N}_{\ge2}\), the \(b\)-adic expansion of a real number is uniquely determined.

Proof: The geometric series implies the claim without requiring any additional conditions.\(\square\)

Theorem: For every \(b \in ]1, 2[\) the \(b\)-adic expansion of any real number \(\ne 0\) is ambiguous.

Proof: Replacing some ones in \(0.{\overline{1}}_{b} > {1}_{b}\) implies the claim.\(\square\)

Definition: The base 2 satisfies the minimality property. Dual numbers are denoted \(d\) followed by the corresponding set. For a constant or variable (here: \(a \in \mathbb{C}\), let \(\acute{a} := a - 1\) and \(\grave{a} := a + 1\). For an expression (here: \(e \in \mathbb{C}^*\)), let \(\hat{e} := 1/e\). Let \(|M|\) be the number of elements in a set \(M\). Let \(c := 2^\hslash\) be for \(\hslash \in \mathbb{N}^{*}\) the largest finite, \(\omega := 2^\ell\) for \(\ell \in \mathbb{N}^{*}\) the largest non-infinite (i.e. inconcrete) and \(\varsigma := 2^\wp\) for \(\wp \in \mathbb{N}^{*}\) the largest infinite real number according to the filling principle for {0, 1} among the dual numbers.

Definition: Let \(d0 := 2^{-\wp}\) be the smallest positive number (minimal distance from 0). Let \(\hslash, \ell\) and \(\wp\) be for \(n \in \mathbb{N}^{*}\) of the form \(x_{\grave{n}} := x_n^2\) for \(x_0 := 2.\) For reasons of clarity, the underlying set will be specified after an interval. Let \({}^{\omega}\mathbb{R} := [-\omega, \omega]\mathbb{R}\) and \({}^{\omega}\mathbb{C} := {}^{\omega}\mathbb{R} + i{}^{\omega}\mathbb{R} \subset \mathbb{C}\). In the following, the notation \({}^{\omega}\) before a real or complex set denotes the intersection with \([-\omega, \omega]\mathbb{R}\) or \({}^{\omega}\mathbb{C}\), implying that the set only contains non-infinite elements. The notation with \({}^{c}\) is defined analogously.

Remark: Sets denoted with \({}^{c}\) are corresponding to the conventional ones without \({}^{c}\). The definition (and thus limitation) of \(c\) resp. \(\omega\) clearly causes these sets to lose the property of being closed. 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. Our construction includes both the conventionally hyperreal and surreal numbers.

Definition: The sum \[p(z)=\sum\limits_{k=0}^{\acute{m}}{{{a}_{k}}{{z}^{k}}}\] where \(z \in \mathbb{C}\) and \(m \in \mathbb{N}^*\) is called an \(m\)-polynomial, if the number of coefficients with e.g. \({a}_{k} \in {}^{c}\mathbb{Z}\) or \({a}_{k} \in {}^{\omega}\mathbb{Z}\) and \(k \in \mathbb{N}_{<m}\) and \({a}_{k} \ne 0\) is finite, otherwise \(m\)-series. Then \(\deg(p) := \acute{m}\) for \({a}_{k} \ne 0\) is called the degree of the polynomial or series \(p\). For the zero polynomial \(p = 0\), we have that \(\deg(p) := -1\). The numbers \(z \in \mathbb{C}\) for which the polynomial or series sums to zero (the zeros of the polynomial or series) are said to be \(m\)-algebraic. The sets of \(m\)-algebraic numbers are denoted \({}^{m}{\mathbb{A}}_{\mathbb{R}}\) in the real case and \({}^{m}{\mathbb{A}}_{\mathbb{C}}\) in the complex case. In the special case \({a}_{\deg(p)} = 1\), we say that the sum is an \(m\)-algebraic integer. The numbers \(z \in \mathbb{C}\) that are not a zero of any \(m\)-polynomial or \(m\)-series are said to be \(m\)-transcendental. When \(m := c\), this gives the conventional notion of transcendental numbers.

Definition: We can write \(m\)-algebraic numbers in the form \({(m, {a}_{k-1}, {a}_{k-2}, ..., {a}_{1}, {a}_{0}; r, i; \#n, Mq; v, p)}_{s}\), where \(r = i = {a}_{0} = 0\) represents the number 0, \(r \in {}^{c}\mathbb{N}^{*} (-{}^{c}\mathbb{N}^{*})\) denotes the \(r\)-th largest (\(|r|\)-th smallest) zero with real part > 0 (< 0); when \(r = 0, i \in {}^{c}\mathbb{N}^{*} (-{}^{c}\mathbb{N}^{*})\) denotes a non-real zero with the \(i\)-th largest (\(|i|\)-th smallest) imaginary part > 0 (< 0), and the other algebraic numbers have analogous notations, with \(r\) taking precedence over \(i\). The value \(\#n\) gives the quantity \(n \in {}^{c}\mathbb{N}^{*}\) of zeros. When at least one \({a}_{j}\) is taken as a variable, \(Mq\) gives the number \(q \in {}^{c}\mathbb{N}\) of repeated zeros. For \(k\)-minimal polynomials or series \(({a}_{0} = 0\) only for the zero polynomial), the sign < is taken as the value of the specification \(s\), in every other case, the sign > is used. The numerical value \(v\) is given up to the precision \(p\).

Remark: This allows the zeros of \(k\)-polynomials or \(k\)-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 \(r, i, \#n, Mq, v, p\), and \(s\) can optionally be omitted (e.g. for rational numbers). The \((|{}^{c}\mathbb{N}|+2)\)-tuple \((0, ..., 0, {a}_{k-1}, ..., {a}_{0}; r, i{)}_{<}\) where the \({a}_{j}\) are finite natural numbers, gives a strict lexical well-ordering of the algebraic numbers.

Examples: The finite numbers \((c, 1, 0, 0, 0, -1{)}_{>}\) are 1, -1, \(i\), and \(-i\). The finite golden ratio \((1 + \sqrt{5})/2\) may be written as \((c, 1, -1, -1; 1, 0; 1.618033, {10}^{-6}{)}_{<}\). The number \(0.\overline{1} = 0.1...1\) with \(\omega\) ones after the point is inconcrete and distinct from the finite number \(\hat{9}\), since 9 \(\times \; 0,\overline{1} = 0.9...9 = 1 - {10}^{-\omega} \ne 1\). It is therefore \(\omega\)-transcendental (cf. Number Theory) and should therefore be written as (\(\omega, 9 \times {10}^{\omega}, 1 - {10}^{\omega})\).

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.\(\square\)

Theorem: An arbitrary mapping \(f: X \rightarrow X\) on an arbitrary set \(X\) is bijective if it is either injective or surjective.

Proof: The claim follows directly from the fact that all (pre-)images are pairwise distinct.\(\square\)

Remark: Note that this theorem does not apply to the successor function \(s\) in \({}^{\omega}\mathbb{N}\), since \(s: {}^{\omega}\mathbb{N} \rightarrow {}^{\omega}\mathbb{N}^{*} \cup \{|{}^{\omega}\mathbb{N}|\}\).

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

Refutation: We have that \(P(\omega, \omega) = 2\omega\grave{\omega} > \omega = \max \; {}^{\omega}\mathbb{N}.\square\)

Remark: Similarly, the Fueter-Pólya conjecture is refuted. If the set \({}^{\omega}\mathbb{N}^{2}\) is replaced by \((m, n) \in {}^{\omega}\mathbb{N}^{2} : m + n \le k \in {}^{\omega}\mathbb{N}\) for \(k(k + 3) = 2\omega\), 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.

Example: For \(x \in {}^{\omega}{\mathbb{R}}_{\ge 1}\), the mapping \(\hat{x}\) maps \({}^{\omega}{\mathbb{R}}_{\ge 1}\) bijectively to the interval \([\hat{\omega}, 1] \subset [0, 1]\), which is bounded on both sides. Therefore, we can analogously map all infinite subintervals of \(\mathbb{R}^{*}\) 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 \(\mathbb{N}\) is a proper subset of \(\mathbb{Q}\). 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 \(|{}^{c}\mathbb{N}|\) and \(|{}^{c}\mathbb{R}|\).

Remark: The symbol \(\infty \gg \varsigma^2\) can be adjoined to the real numbers. It may be used in calculations like a constant. Since division by 0 is not defined in calculations, we can simplify things e.g. by replacing \(\pm0\) by \(\pm\hat{\infty}\) wherever it makes sense to do so, depending on which direction is relevant to the present case, and calculate with \(\infty\) uniquely and without contradiction. 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 \(\pi\) as the area or half of the circumference of the unit circle. We define Euler's number \(e\) as the solution of the equation \({x}^{i\pi} = -1\). 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: The alternative definition of \(e := {(1 + \hat{c})}^{c}\) is \(\mathcal{O}(\hat{c})\) 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.

Remark: Let \(m \in {}^{c}\mathbb{N}\) be the maximum polynomial degree and \(n \in {}^{c}\mathbb{N}\) 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 \(k \in {}^{c}\mathbb{N}_{\le m}\) 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 \({a}_{0} \ne 0\).

Counting theorem for algebraic numbers: The number \(\mathbb{A}(m, n)\) of algebraic numbers (of polynomial or series degree \(m\) and thus in general) asymptotically satisfies the equation\[\mathbb{A}(m, n) = \widehat{\zeta(\grave{m})}\,z(m){{(2n+1)}^{m}}\left( n+\mathcal{O}\text{(ln }n) \right),\]where \(\zeta\) is the Riemann zeta function and \(z(m)\) is the (average) number of zeros of a polynomial or series.

Proof: The case \(m = 1\) is proven in [455] and the error term \(\mathcal{O}(n \, \ln \, n)\) is required when estimating the number of rational numbers with the Euler \(\varphi\)-function by \(4\sum\limits_{k=1}^{n}{\varphi (k)}-1\). For \(m > 1\), the divisibility conditions neither change the error term \(\mathcal{O}(\ln \, n)\) nor the leading term. The factor of \(1/\zeta(\grave{m})\) eliminates all polynomials or series such that gcd\(({a}_{0}, {a}_{1}, ... , {a}_{m}) \ne 1\). To remove repeat prime numbers \(p\), we must multiply the number of polynomials or series by \((1 - {p}^{-\grave{m}})\). Taking the product over all prime numbers and developing the factors into geometric series gives the \(1/\zeta(\grave{m})\) after multiplying out. If precisely one coefficient is 0, \(\zeta(\grave{m})\) can be replaced by \(\zeta(m)\). 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.\(\square\)

Examples: For \(m = 1\), we obtain \(3(\hat{\iota}n)^{2}+\mathcal{O}(n \text{ ln }n)\) rational solutions. For \(m = 2\), we obtain \({4.5n}^{3}/\zeta(3) + \mathcal{O}({n}^{2}\ln \, n)\) 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 \(z(m){(2n+1)}^{\acute{m}}(2n + \mathcal{O}(\ln \, n))\) algebraic integer solutions.

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 \(\hat{\iota} \, \ln \, m + \mathcal{O}(1)\) according to (Kac, Mark: On the Average Number of Real Roots of a Random Algebraic Equation. II.; Proc. London Math. Soc. 50; 1949; 390 - 408).

Examples: For \(m = n = \acute{c}\), omitting the \({}^{c}\) before each set, we obtain in the real case\[\left| {{\mathbb{A}}_{\mathbb{R}}} \right|=\hat{\iota }{{\left| \mathbb{Z} \right|}^{\left| \mathbb{N}* \right|}}\text{ln}\left| \mathbb{N} \right|\left( \left| \mathbb{N} \right|+\mathcal{O}\left( \text{ln}\left| \mathbb{N} \right| \right) \right)\]and in the complex case\[\left| {{\mathbb{A}}_{\mathbb{C}}} \right|=\hat{2}{{\left| \mathbb{Z} \right|}^{\left| \mathbb{N} \right|}}\left( \left| \mathbb{N} \right|+\mathcal{O}\left( \text{ln}\left| \mathbb{N} \right| \right) \right).\]Remark: We can only (uniquely) determine the number of elements of an inconcrete or infinite set if we know its construction. Then we can compare the set to \({}^{(\omega)}\mathbb{N}\), which may be taken as a basis thanks to its simple construction. If there are multiple possible constructions, we take or specify the most plausible, i.e. the one that best reflects the non-finiteness of the set for the purpose of differentiating between the possible cases.

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

Proof: Let \(a \in {}^{c}{\mathbb{R}}_{>1}\). Then \(\hat{c} m \le 1 < a\) for all \(m \in {}^{c}\mathbb{N}.\square\)

Archimedes' theorem: There exists \(m \in {}^{c}\mathbb{N}\) such that \(d m > a\) if and only if \(d c > a\) whenever \(a > d\) for \(a, d \in {\mathbb{R}}_{>0}\).

Proof: Among all \(m \in {}^{c}\mathbb{N}, c\) is the largest one.\(\square\)

Definition: We replace \(\mathbb{K}\) either by \(\mathbb{R}\) or \(\mathbb{C}\). Two different points \(x\) and \(y\) in a subset \(M \subseteq \mathbb{K}^{n}\) where \(n \in \mathbb{N}^{*}\) are said to be neighbours if \(||x - y|| \le \max \, \{||x - z||, ||y - z||\}\) holds for all points \(z \in M\), where \(||\cdot||\) denotes the Euclidean norm. The subsets of \(\mathbb{K}^{n}\) such that all neighbouring points have the minimum distance \(d0\) are said to be gapless.

Definition: A non-empty subset \(M \subseteq \mathbb{R}\) is said to be \(h\)-homogeneous if the minimum distance between any two of its points is \(h \in \mathbb{R}_{>0}\). We denote this by \(h\)-\(M\). An \(n\)-dimensional subset \(M \subseteq \mathbb{R}^{n}\) with \(n \in \mathbb{N}^{*}\) is said to be \(h\)-homogeneous if it is \(h\)-homogeneous in each dimension. We define \(h\)-homogeneity analogously for subsets of \(\mathbb{C}^{n}\). A subset \(M \subseteq \mathbb{K}^{n}\) is said to be dense in \(\mathbb{K}^{n}\) if there is a point \(y \in M\) for every \(x \in \mathbb{K}^{n}\) 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 \({}^{c}\mathbb{A}_{\mathbb{Q}} \subset {}^{c}\mathbb{Q}\) is true and also the inhomogeneity of \({}^{c}\mathbb{A}_{\mathbb{C}} \subset {}^{c}\mathbb{C}\). The maximum number of leading and also fractional digits of elements of \(\hat{c}\)-\({}^{c}\mathbb{R}, \hat{\omega}\)-\({}^{\omega}\mathbb{R}\) and \((\hat{\varsigma}\)-) \(\mathbb{R}\) is given by \(\hslash, \ell\) or \(\wp\).

Fundamental theorem of set theory: The set \((d\mathbb{R} = \mathbb{Q} =) \; \mathbb{R}\) is a maximal, totally ordered, closed, continuous and \(d0\)-homogeneous field giving \(|\mathbb{R}| = 2 {\varsigma}^{2} + 1\).

Proof: We cannot distinguish \(h\)-homogenised elements arbitrarily precisely.\(\square\)

Remark: Therefore, irrational numbers do not exist and \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\) and \(\mathbb{C}\) are \(h\)-homogeneous.

© 2002-2018 by Boris Haase


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