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Set Theory

Set Theory

Definition: Let two entities be identical if they shall not be distinguished. The entirety of non-identical entities is called set \(S\) of elements such that \(|S|\) is their number \(n\), \(\emptyset\) denotes the empty set. If \(S \ne \emptyset\) can be emptied by successively removing the half of the remaining elements, which is rounded up until \(S\) is empty, then \(n\) is finite, otherwise infinite. Let \(\nu\) the largest finite real number, \(\omega\) in between the largest mid-finite and \(\varsigma\) the largest infinite one according to the filling principle for \(\{0, 1\}\) among the dual numbers.\(\triangle\)

Remark: The existence of actual or only potentially infinite sets remains open, since the transcendence of the infinite allows here no proof. Conventionally, sufficiently well-understood axioms exist that define the real numbers as a totally ordered field and the complex numbers as a field with the imaginary unit \(i\). Analogously, addition, multiplication, and their inverse operations may be extended to the largest and by definition closed field extensions \(\mathbb{R}\), and \(\mathbb{C} := \mathbb{R} + i\mathbb{R}\) (with further operations).

Definition: The set of natural numbers \(\mathbb{N} := \mathbb{N}^{*} \cup \{0\}\) is defined as all numbers obtained by successively adding 1 to 0. The set of prime numbers \(\mathbb{P}\) is defined by excluding all composite numbers from \(\mathbb{N}_{\ge 2}\). 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 by adding the set of fractions with numerator from \(\mathbb{Z}^{*}\) and denominator from \(\mathbb{N}^{*}\) to \(\mathbb{Z}\). The set of complex-rational numbers is \(\mathbb{Q} + i\mathbb{Q}\). Let d0 \(:= \hat{\varsigma} = \min \mathbb{R}_{>0} \) the minimal distance from 0.\(\triangle\)

Minimality theorem: For the unique \(b\)-adic expansion of every \(r \in \mathbb{R}\), min \(\{b \in \mathbb{R}_{>1}\} = 2\) holds.

Proof: Focusing on the positions after the floating point, the geometric series yields the claim.\(\square\)

Definition: Decrement and increment of \(a \in \mathbb{C}\) are given by \(\acute{a} := a - 1\) and \(\grave{a} := a + 1\). The reciprocal of \(u \in \mathbb{C}^*\) is \(\hat{u} := 1/u\). Let \({}^{\omega}\mathbb{R} := [-\omega, \omega]\) and \({}^{\omega}\mathbb{C} := {}^{\omega}\mathbb{R} + i{}^{\omega}\mathbb{R} \subset \mathbb{C}\). If \({}^{\omega}\) precedes a real or complex set, it denotes the intersection with \([-\omega, \omega]\) or \({}^{\omega}\mathbb{C}\). Between the finite and the infinite numbers, there are the mid-finite real numbers \({\mathbb{M}}_{\mathbb{R}} := {}^{\omega}{\mathbb{R}} \setminus {}^{\nu}{\mathbb{R}}\) where the complex yields \({\mathbb{M}}_{\mathbb{C}} := {\mathbb{M}}_{\mathbb{R}} + i{\mathbb{M}}_{\mathbb{R}}.\) For real sets, the notation may omit \({\mathbb{R}}.\triangle\)

Remark: Minimality theorem and digital computer explain the choice of 2. The latter sets only contain non-infinite elements. The notation with \({}^{\nu}\) is defined analogously for \([-\nu, \nu]\) or \({}^{\nu}\mathbb{C}\). Sets denoted with \({}^{\nu}\) are corresponding to the conventional ones without \({}^{\nu}\). The definition and thus limitation of \(\nu\) resp. \(\omega\) clearly causes these sets to lose the property of being closed. The almost arbitrary definitions of the largest finite and (non-)infinite real number have no convincing alternatives.

Remark: An abrupt transition between finite and infinite numbers is hard to justify. Only the unique construction of a mid- or infinite set allows to determine the number of its elements and to relate it to \({}^{\omega}\mathbb{N}\) as a basis thanks to its simple construction. If there are multiple possible constructions, the most plausible one is selected. The existing set \(\mathbb{W}\) of all sets (worlds) cannot be changed. \(\mathbb{R}\) includes both the conventionally hyperreal and surreal numbers.

Definition: Let \(\in\) be irreflexive and asymmetric, whereas \(\subseteq\) is partial order. Two sets are equal if and only if they contain the same elements (extensionality). The set \(Y\) is called union of the set \(X\) if it contains exactly the elements of the elements of \(X\) as elements. Let \(\mathcal{P}(X) := \{Y : Y \subseteq X\}\) be the power set of the set \(X\). Every number from \(\mathbb{C}^{*}\) is said to be infinitesimal if the absolute value of its reciprocal is infinite.\(\triangle\)

Inclusion theorem: Neither a nonempty set contains itself or its power set, nor it admits a bijection with one of its subsets.

Proof: Every set differs from its elements, since it comprises the latter. Thus \(\emptyset \ne \{\emptyset\}\). Its relative complement shows the elements with missing partner element for the bijection.\(\square\)

Conclusion: In particular, this 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 as the successor function \(s: {}^{\omega}\mathbb{N} \rightarrow {}^{\omega}\mathbb{N}^{*} \cup \{\grave{\omega}\}\) shows for \({}^{\omega}\mathbb{N}\). This contradicts Hilbert's hotel. Because there are infinitely many sets whose number of elements lie between \(|{}^{\nu}\mathbb{N}|\) and \(|{}^{\nu}\mathbb{R}|\), the continuum hypothesis is wrong, too.

Claim: The Cantor polynomial \(P(m, n) := ({(m + n)}^{2} + 3m + n)/2\) bijectively maps \({}^{\omega}\mathbb{N}^{2}\) to \({}^{\omega}\mathbb{N}\).

Refutation: It holds 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.

Definition: The fact that there are no cyclic sequences of sets, each containing one as an element in the previous one, is called cycle freedom. The fact that a set \(X\) be transformed into a set by uniquely replacing each element of \(X\) by an arbitrary set is called replaceability. The set \(Y\) is called selection of pairwise disjoint nonempty sets from a set \(X\) if it contains exactly one element from each element of \(X\) (postulation of selectability). The numbers \(\nu, \omega\) and \(\varsigma\) have for \(n \in \mathbb{N}\) the form \(2^n.\triangle\)

Foundation theorem: Only the postulation of the axiom of foundation that every nonempty subset \(X \subseteq Y\) contains an element \(x_0\) such that \(X\) and \(x_0\) are disjoint guarantees cycle freedom.

Proof: Set \(X := \{x_m : x_0 := \{\emptyset\}, x_{\omega} := \{x_1\}\) and \(x_{\acute{n}} := \{x_n\}\) for \(m \in {}^{\omega}\mathbb{N}\) and \(n \in {}^{\omega}\mathbb{N}_{\ge 2}\}.\square\)

Remark: Setting \(x_{\omega} := \{x_0\}\) instead of \(x_{\omega} := \{x_1\}\), \(X\) becomes an infinite chain. All above definitions determine the set theory represented here, which does not require proper classes.

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 \({a}_{k} \in {}^{\nu}\mathbb{Z}\) or \({a}_{k} \in {}^{\omega}\mathbb{Z}\) where \(k \in \mathbb{N}_{<m}\) 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 \(p = 0\), \(\deg(p) := -1\) holds. The numbers \(z \in \mathbb{C}\) setting \(p(z)\) are called zeros and to be \(m\)-algebraic. The corresponding sets are denoted \({}^{m}{\mathbb{A}}_{\mathbb{R}}\) in the real case and \({}^{m}{\mathbb{A}}_{\mathbb{C}}\) in the complex one. Sets of dual numbers begin with \(d.\triangle\)

Definition: The special case \({a}_{\deg(p)} = 1\) yields \(m\)-algebraic integers. The numbers \(z \in \mathbb{C}\) that are neither a zero of any \(m\)-polynomial nor of any \(m\)-series are said to be \(m\)-transcendental. The corresponding sets are denoted \({}^{m}{\mathbb{T}}_{\mathbb{R}}\) in the real case and \({}^{m}{\mathbb{T}}_{\mathbb{C}} := ({}^{m}{\mathbb{T}}_{\mathbb{R}} + {i}^{m}\mathbb{R}) \cup ({}^{m}\mathbb{R} + {i}^{m}{\mathbb{T}}_{\mathbb{R}})\) in the complex one. When \(m := \nu\), it is spoken of transcendental numbers.\(\triangle\)

Definition: The possibly misleading term of countability should not be used. Let \(\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}\). 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.\(\triangle\)

Definition: A real set \(M \ne \emptyset\) is said to be \(h\)-homogeneous if the minimum distance between any two of its points is \(h \in \mathbb{R}_{>0}\). This is denoted 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. Analogously, \(h\)-homogeneity is defined 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.\(\triangle\)

Remark: To \(h\)-homogenise a set, move \(h\) away from the origin in each dimension and round elements in between up or down to the nearest \(h\)-homogeneous elements. Moreover \({}^{\nu}\mathbb{A}_{\mathbb{Q}} \subset {}^{\nu}\mathbb{Q}\) is true and also the inhomogeneity of \({}^{\nu}\mathbb{A}_{\mathbb{C}} \subset {}^{\nu}\mathbb{C}\). The maximum number of leading and also fractional digits of elements of \(\hat{\nu}\)-\({}^{\nu}\mathbb{R}, \hat{\omega}\)-\({}^{\omega}\mathbb{R}\) and \((\hat{\varsigma}\)-) \(\mathbb{R}\) is given by the logarithms to base 2 (see Nonstandard Analysis) \({_2}\nu, {_2}\omega\) or \({_2}\varsigma\).

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

Proof: Distinguishing arbitrarily precisely \(h\)-homogenised elements is not possible.\(\square\)

Remark: Therefore, irrational numbers do not exist and \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\) as well as \(\mathbb{C}\) are \(h\)-homogeneous. Since rational numbers can have at the most a periodic fractional expansion and can therefore be reconstructed unambiguously, \(h\)-homogeneity does not represent a significant restriction.

© 2002-2018 by Boris Haase

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