Set Theory | Boris Haase's Homepage

Definition: An entity is everything that is distinguishable as a being. A set $S$ is an entirety of entities, called elements, whose number is denoted by $|S|$. Only the empty set $\emptyset$ contains no elements. A set $S \ne \emptyset$ is called finite (otherwise infinite) if and only if it can be completely divided into subsets, where each subset contains the rounded half of the remaining elements. Let $\nu$ the largest finite real number, $\omega$ in between the largest mid-finite and $\iota$ the smallest positive one.$\triangle$

Conventionally, sufficiently well-understood axioms exist that define the real numbers as a linearly ordered field and the complex numbers as a field with the imaginary unit $\underline{1}$ (read as “comp 1”). Analogously, addition, multiplication, and their inverse operations may be extended to the largest and by definition closed field extensions $\mathbb{R}$, and $\mathbb{C} := \mathbb{R} + \underline{\mathbb{R}}$ (with further operations like exponentiation). Every number from $\mathbb{C}^{*}$ is said to be infinitesimal if the absolute value of its reciprocal is infinite.

Presenting Peano axioms and field axioms is deliberately omitted here to remain clear and to emphasise the new content. Two numbers must have a minimum distance, since all different numbers of $\mathbb{R}$ are separated by a distance. Assuming the opposite of the minimum not being fixed leads to the contradiction that every number from $\mathbb{R}$ must have at least one nearest neighbour as a number, which is itself in $\mathbb{R}$ (cf. the isomorphism above).

The set $\mathbb{R}$ of all real numbers is isomorphic to a set of (hyper-) natural or integer numbers. It has both a fixed minimum element and a fixed maximum element, since $\mathbb{R}$ is viewed holistically and completely, from which its closure is obtained together with the field axioms. Otherwise, the former theory would have to be repeatedly adapted to the circumstances. Numbers noted as sequences are plain unwieldy¹.

Cantor regarded many sets as of the same cardinal number. His distinction between merely countable and uncountable sets is not differentiated enough. If instead bijections are correctly considered, this gives a different picture. The fuss made so far over ordinal and cardinal numbers is omitted because there is an easier way: the consistent extension of the real numbers.

Definition: Successively adding 1 to 0 gives the set of natural numbers $\mathbb{N} := \mathbb{N}^{*} \cup \{0\}$. Numbers from $\underline{\mathbb{R}}^{*}$ are called purely complex. Excluding all composite numbers from $\mathbb{N}_{\ge 2}$ gives the set of prime numbers $\mathbb{P}$. The set of integers $\mathbb{Z}$ is given by $\mathbb{Z} := \mathbb{N} \cup -\mathbb{N}^*$ and the set of real numbers $\mathbb{R}$ by the set of fractions with numerator from $\mathbb{Z}$ and denominator from $\mathbb{N}^{*}$. The minimal distance to 0 is $\iota$. $\mathbb{R}$ includes both the conventionally hyperreal and surreal numbers. The numbers $\nu, \omega$ and $1/\iota$ have the form $2^n$ for $n \in \mathbb{N}.\triangle$

Definition: Let $\mathbb{B} = \{0, 1\}$ the Boolean domain. Decrement and increment of $a \in \mathbb{C}$ are given by $\acute{a} := a – 1$ and $\grave{a} := a + 1$ and the reciprocal of $u \in \mathbb{C}^*$ is $\tilde{u} := 1/u$. They are pronounced “$a$ dec”, “$a$ inc” and “turn $u$”. Furthermore, $\hat{a} := 2a, \check{a} := a/2$ and $a$- $:= -a$ are pronounced “a hat”, “half a” and “a neg”. The logarithm of $a \in {}^{\omega}\mathbb{C} \setminus {}^{\omega}\mathbb{R}_{\le 0}$ and base $b \in {}^{\nu}\mathbb{R}_{>0}$ is denoted by ${_b}{a}$ and pronounced “$b$ log $a$”. If $\sigma \in \{\nu, \omega\}$ and $S^* = S^{*-1}$, let ${}^\sigma S := S \cap [-\sigma, \sigma]$ for $S \subseteq \mathbb{R}, {}^\sigma \mathbb{C} := {}^\sigma \mathbb{R} + {}^\sigma \underline{\mathbb{R}} \subset \mathbb{C}$ and ${}^\sigma S := S \cap {}^\sigma \mathbb{C}$ for $S \subseteq \mathbb{C}.\triangle$

Definition: Concerning all $s \in {_\tau^\sigma}S \subset \mathbb{C}$, the maximum value of their numerators is specified by $\sigma$ and that of their denominators by $\tau$. The sets ${\mathbb{M}}_{\mathbb{R}} := {}^{\omega}{\mathbb{R}} \setminus {}^{\nu}{\mathbb{R}}$ and ${\mathbb{M}}_{\mathbb{C}} := {\mathbb{M}}_{\mathbb{R}} + \underline{\mathbb{M}}_{\mathbb{R}}$ result in the mid-finite numbers. For real sets, subscript $_{\mathbb{R}}$ may be omitted in the notation. 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 $X.\triangle$

Remark: For subsets of $S$, this holds analogously. For $T := {}^\sigma S \notin \{\emptyset, \{0\}\}$, it holds that $T+T \nsubseteq T$ or even $T \cdot T \nsubseteq T$. Sets with leading ${}^{\omega}$ only contain non-infinite elements. Sets ${}^{\nu}S$ are related to the conventional sets $S$. Mid-finite sets close the gap between finite and infinite ones.

Extreme theorem: Every linear order has exactly both a maximum and a minimum element, since all other constellations lead to a contradiction to totality or to that mentioned above.$\square$

Remark: 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 basis thanks to its simple construction. If there are multiple possible constructions, the most plausible one is selected. The relation $\in$ is irreflexive and asymmetric, whereas $\subseteq$ is partial order.

Definition: The sum $p(z)={\LARGE{\textbf{+}}}_{k=0}^{\acute{m}}{{{a}_{k}}{{z}^{k}}}$ for $z \in \mathbb{C}$ 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, else $m$-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 $z \in \mathbb{C}$ setting $p(z) = 0$ are called zeros and are each an $m$-AN.$\triangle$

Definition: Let min $|p(z)| = |p(\alpha)|$ for the minimal polynomial $p(z)$ of $\alpha \in \mathbb{C}$ where $0 \ne p(\alpha) \approx 0$. The corresponding sets are denoted $A := {}^{m}{\mathbb{A}}_{\mathbb{R}}$ in the real case and ${}^{m}{\mathbb{R}}_{\mathbb{C}} \supset {}^{m}{\mathbb{A}}_{\mathbb{C}} \supset {}_m^{m}{\mathbb{R}}_{\mathbb{C}} \supset {}^{m}{\mathbb{Z}}_{\mathbb{C}} \supset {}^{m}{\mathbb{N}}_{\mathbb{C}}$ in the complex one. For ${a}_{\deg(p)} = 1$, $m$-algebraic integers are given. The notation for $m$-ANs is ${(m, {a}_{\acute{k}}, {a}_{k-2}, …, {a}_{1}, {a}_{0}; r, i; \#n, \&q; v, p)}_{s}$. All $k$-minimal polynomials have the sign < as specification $s$, all $k$-minimal series have >.

Definition: For $r \in {}^{\nu}\mathbb{N}^{*} (-{}^{\nu}\mathbb{N}^{*})$ exists the $r$-th largest ($|r|$-th smallest) zero with real part > 0 (< 0), where $r = 0, i \in {}^{\nu}\mathbb{N}^{*} (-{}^{\nu}\mathbb{N}^{*})$ denotes a non-real zero with the $i$-th largest ($|i|$-th smallest) imaginary part > 0 (< 0), and the other ANs have analogous notations. The value $\#n$ gives the quantity $n \in {}^{\nu}\mathbb{N}^{*}$ of zeros. When at least one ${a}_{j}$ is taken as a variable, $\&q$ gives the number $q \in {}^{\nu}\mathbb{N}$ of repeated zeros.$\triangle$

Remark: Here $r$ takes precedence over $i$ and $r = i = {a}_{0} = 0$ represents the number 0. The numerical value $v$ has the precision $p$. Not distinguishing between repeated zeros allows the zeros of $k$-polynomials or $k$-series with integer coefficients to be endowed with a strict total ordering. The information $r, i, \#n, \&q, v, p$, and $s$ may optionally be omitted as e.g. for fractions. The $(\nu+2)$-tuple $(0, …, 0, {a}_{\acute{k}}, …, {a}_{0}; r, i{)}_{<}$ where every ${a}_{j} \in {}^{\nu}\mathbb{N}$ gives a strict lexical well-ordering of the ANs.

Examples: The numbers $(\nu; 1, 0, 0, 0, -1{)}_{<}$ are $1, -1, \underline{1}$, and $-\underline{1}$. The golden ratio $\Phi := \check{1} + {5}^{\tilde{2}}/2$ may be written as $(\nu; 1, -1, -1; 1, 0; 1.618033, {10}^{-6}{)}_{<}$. Precision $p = {10}^{-\omega}$ implies $\tilde{9} \ne 0.\overline{1} = 0.1…1$, since $9\,\times\,0.\overline{1} = 0.9…9 = 1 – {10}^{-\omega} \ne 1$. Thus $0.\overline{1}$ is not $\omega$-algebraic and may be written as ($\omega, 9 \times {10}^{\omega}, 1 – {10}^{\omega})$.

Remark: Let $m \in {}^{\nu}\mathbb{N}$ be the maximum polynomial degree and $n \in {}^{\nu}\mathbb{N}$ the maximum absolute value that the integer coefficients ${a}_{k}$ can take of the polynomials ${a}_{m}{x}^{m} + {a}_{\acute{m}}{x}^{\acute{m}} + … + {a}_{1}x + {a}_{0}$ with $k \in {}^{\nu}\mathbb{N}_{\le m}$. This makes sense due to the symmetry of the ${a}_{k}$. The number of ANs 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 ${a}_{0} \ne 0$.

Inclusion theorem: Every set $S \ne \emptyset$ falsifies $S \in S$, $\mathcal{P}(S) \subseteq S$ and any isomorphism $T \cong S$ for $T \subset S$.

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: Especially, this contradicts Dedekind-infiniteness and Cantor’s first diagonal argument, since $\mathbb{N}$ is a proper subset of $\mathbb{R}$. Putting numbers excluded from the beginning to the end refutes the second one. The same is true for the Cantor, the Russell-Zermelo and 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.

Definition: Cycle freedom denotes the absence of cyclic sequences of sets, each containing one as an element in the previous one. Replaceability denotes the possible transition from a set $X$ by uniquely replacing each element of $X$ by an arbitrary set. If set $Y$ contains exactly one element from each element of $X$ (postulation of selectability), it is called selection of pairwise disjoint nonempty sets from a set $X.\triangle$

Lemma: Because of $\tilde{\nu}m \le 1 \le a$ for all $m \in {}^{\nu}\mathbb{N}$ and $a \in {}^{\nu}{\mathbb{R}}_{\ge 1}$, the Archimedean axiom is invalid.$\square$

Archimedes’ theorem: There exists $m \in {}^{\nu}\mathbb{N}$ such that $a < bm$ if and only if $a < b\nu$ whenever $a > b$ for $a, b \in {\mathbb{R}}_{>0}$, since $\nu = \max {}^{\nu}\mathbb{N}$ holds.$\square$

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 $P(\omega, \omega) = \hat{\omega}\grave{\omega} >\omega = \max \; {}^{\omega}\mathbb{N}.\square$

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 $k(k + 3) = \hat{\omega}$, the claim holds. Since infinitely many sets exist whose number of elements lie between $|{}^{\nu}\mathbb{N}|$ and $|{}^{\nu}\mathbb{R}|$, the continuum hypothesis is wrong, too: By mapping too many (infinite) sets onto each other, they cannot be longer told apart.

Foundation theorem: Putting $X := \{x_m : x_0 := \{\emptyset\}, x_{\omega} := \{x_1\}$ and $x_{\acute{n}} := \{x_n\}$ for $m \in {}^{\omega}\mathbb{N}$ and $n \in {}^{\omega}\mathbb{N}_{\ge 2}\}$, only postulating the axiom of foundation guarantees cycle freedom that every nonempty subset $X \subseteq Y$ contains an element $x_0$ such that $X$ and $x_0$ are disjoint.$\square$

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 classes² since only meaningful sets are considered. The existing set $\mathbb{W}$ of all worlds cannot be changed, because otherwise it would contradict completeness. However, this does not mean that changes in the worlds do not exist or that everything is determined. It helps to look at time spatially.

Definition: 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. If all neighbouring points of $M \subseteq \mathbb{K}^{n}$ have the minimal distance $h \in \mathbb{R}_{>0}$, $M$ is said to be $h$-homogeneous and is denoted by $h$-$M$. The case $h = \iota$ makes $M$ gapless. The isomorphism $\mathbb{C}^{n} \cong \mathbb{R}^{\hat{n}}$ holds.$\triangle$

Definition: Let $u{\upharpoonleft}_n := (u, …, u)^T \in{}^{\omega}\mathbb{K}^{n}$. Read $u{\upharpoonleft}_n$ as “u rep 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 $||x – y|| = \iota$. For the point-symmetric ${\dot{\mathbb{K}}}^n$ related to $\mathbb{K}^{n}$, an $n$-cube with edge length $\sigma$ is given by ${}^{\sigma}\mathbb{R}^{n} := {\LARGE{\textbf{$\times$}}}_{k=1}^n{{}^{\sigma}\mathbb{R}}$ and an $n$-ball by ${}^{\sigma}\dot{\mathbb{R}}^n$ with radius $\sigma$. Let $\mathbb{U}$ the fuzzy set, $\mathbb{Q}$ the set of quaternions, $\mathbb{O}$ the set of octonions and $\mathbb{S}$ the set of sedenions.$\triangle$

Remark: To $h$-homogenise a set, move $h$ away from the origin in each dimension. Round elements in between to the nearest $h$-homogeneous elements. The maximum number of leading and also fractional digits of elements of $\tilde{\nu}$-${}^{\nu}\mathbb{R}, \tilde{\omega}$-${}^{\omega}\mathbb{R}$ and $(\iota$-) $\mathbb{R}$ is given by the logarithms to base 2 (see Nonstandard Analysis) ${_2}\nu, {_2}\omega$ or ${_2}\tilde{\iota}$. The following theorem solves the Erdős-Ulam problem and reveals irrational numbers to be non-existent, and $\mathbb{N}, \mathbb{Z}, \mathbb{R}$ as well as $\mathbb{C}$ to be $h$-homogeneous.

Fundamental theorem of set theory: The set $\mathbb{R}$ is a maximal, linearly ordered, closed, continuous and $\iota$-homogeneous field giving $|\mathbb{R}| = 2 {\tilde{\iota}}^{2} + 1$, since the distance $\iota$ guarantees continuity, although the exact value of $\iota$ is undetermined, and $\mathbb{R}$ can be “glued together” by identifying min $\mathbb{R}$ and max $\mathbb{R}.\square$

Remark: Glue subsets of $\mathbb{K}^{n}$ analogously, which is then closed for certain calculations. Since reconstructing real numbers is unambiguous with necessarily periodic fractional expansion, $h$-homogeneity does not restrict significantly. A complete homogeneity (i. e. in all directions) of higher-dimensional real spaces is unfortunately impossible, as a simple consideration of circles or balls shows ‒ also when one-dimensional spaces are rolled up.

Conclusion: A linear order can be defined for every non-empty set.$\square$

Remark: Unlike the conventional way³, immediate predecessors and successors can be specified. Vividly, this is a threading of pearls into a chain where the number of dimensions does not matter. Individual points from $\mathbb{R}^n$ can be linearly ordered with $n \in {}^{\omega}\mathbb{N}^*$, for example, dimension by dimension: The total chain length is then $\iota|\mathbb{R}|^n$. Another linear order is the spiral order starting at $0{\upharpoonright}_n$ according to the Euclidean norm for individual points.

Remark: Only malicious gossip has it that is wrong what is not trivially true in Cantor’s set theory. What is certain, however, is that much of the latter cannot be maintained so far, but still contains enough interesting ideas to be pursued. A more detailed evaluation is omitted here for reasons of fairness.

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References

cf. Landers, Dieter; Rogge, Lothar: Nichtstandard Analysis; 1. Aufl.; 1994; Springer; Berlin, p. 15 f.
cf. Deiser, Oliver: Einführung in die Mengenlehre; 1. Aufl.; 2002; Springer; Berlin, p. 117 f.
cf. Schmidt, Gunther; Ströhlein, Thomas: Relationen und Graphen; 1. Aufl.; 1989; Springer; Berlin, p. 33.