# Homepage of Boris Haase

## #50: Completion Set Theory on 10.11.2018

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.