Set Theory

Set Theory
Set Theory

Definition: Forgoing to distinguish two entities makes them identical. The entirety of non-identical entities is called set \(S\) of elements such that \(|S|\) is their number \(n\). Symbol \(\emptyset\) denotes the empty set, which contains no elements. If \(S \ne \emptyset\) can be emptied by successively removing 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.\(\triangle\)

Preliminary considerations: Referencing to philosophical terms is justified since mathematics cannot be explained in terms of metalinguistics alone and abstract words provide the most general statements according to the principle of scientificity. The word successive here contains the physical idea that there is a measurable or in some form perceptible time span between the points in time to be considered, and thus implicitly assumes the Planck time of approx. \(10^{-43}\) seconds.

The reproach of lacking general validity due to the inclusion of physical facts must be countered that in this world arbitrarily small periods of time or genuinely instant events have no meaning, since they do not exist in practice, but are at most allowed for theoretical considerations. Due to the structurally similar quantities of space and time combined in space-time, the statements below concerning space apply to time, too.

Specifying neither finite nor infinite numbers requires midfinite ones. Accepting an abrupt transition here would be difficult to justify, since finite and infinite represent two opposites that are not easy to reconcile. The existence of actual or only potentially infinite sets remains open, since the transcendence of the infinite fails to provide a proof. What does the question make plausible: If something is conceivable, can its existence be impossible?

Are there contradictions or inconsistencies if everything were only finite? One consequence would be the compulsory repetition of everything in the finite case at a certain point in time. From the almost “wasteful” size or expansion of the universe, it can be concluded that there is an availability that has no recognisable limits. Even so, such inferences remain weaker than abduction due to the practical unavailability of infinity.

If a finite line is broken down into an infinite number of parts, the infinite is finitely limited. If, in addition, the number of parts of both infinities is the same, there is mathematically an isomorphism: The enlargement of the infinitely small parts of the finite segment to finite ones results in infinity in the conventional sense in relation to the whole. That easily deduces a bijection in the mathematical sense.

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 \(i\). 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 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).

Definition: Successively adding 1 to 0 gives the set of natural numbers \(\mathbb{N} := \mathbb{N}^{*} \cup \{0\}\). 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 rationals \(\mathbb{Q}\) by the set of fractions with numerator from \(\mathbb{Z}\) and denominator from \(\mathbb{N}^{*}\). The minimal distance to 0 is d0\(:= \hat{\varsigma} = \min \mathbb{R}_{>0} \). \(\mathbb{R}\) includes both the conventionally hyperreal and surreal numbers. The numbers \(\nu, \omega\) and \(\varsigma\) have the form \(2^n\) for \(n \in \mathbb{N}.\triangle\)

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.\(\square\)

Remark: For \(b \in \; ]1, 2[\), the development is no longer clear, as two digits (0 and 1) intended for the dual representation must be used: Restricting to one digit only makes sense for the base \(b = 1\). For \(b > 2\), uniqueness can be established because of \((1 – b^{-n})/(1 – b^{-1}) – 1 < 1\) where \(n \in \mathbb{N}\) denotes the number of decimal places of \(r\). Minimality theorem and digital computers, which are working most often within a binary system, explain the choice of 2 as basis.

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, subscript \(\mathbb{R}\) may be omitted in the notation.\(\triangle\)

Remark: The square (e. g. given by \({}^{\omega}\mathbb{C}\)) is preferred to the circle, since the Cartesian product is more logical and easier to represent, although its diagonals would be its longest straight line. 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.

Extreme theorem: Every linear order has exactly both: a maximum and a minimum element.

Proof: 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 existing set \(\mathbb{W}\) of all worlds cannot be changed, because otherwise there would be a contradiction in terms of completeness. The approach of this homepage is based closely on ISO 80000-2: 2019 (quantities and units – mathematics).

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 \(X.\triangle\)

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: Especially, 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 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. Since infinitely many sets exist 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: 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 the 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 \(X.\triangle\)

Foundation theorem: Only postulating 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 \(p(z)=\sum\limits_{k=0}^{\acute{m}}{{{a}_{k}}{{z}^{k}}}\) 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 numbers \(z \in \mathbb{C}\) setting \(p(z) = 0\) are called zeros and are \(m\)-algebraic. The corresponding sets are denoted \({}^{m}{\mathbb{A}}_{\mathbb{R}}\) in the real case and \({}^{m}{\mathbb{A}}_{\mathbb{C}} \supset {}^{m}{\mathbb{Q}}_{\mathbb{C}} \supset {}^{m}{\mathbb{Z}}_{\mathbb{C}} \supset {}^{m}{\mathbb{N}}_{\mathbb{C}}\) in the complex one.\(\triangle\)

Definition: For \({a}_{\deg(p)} = 1\), \(m\)-algebraic integers are given. 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 “\(b\) in \(a\)”. 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. Conventional transcendence is given by \(m := \nu.\triangle\)

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 minimal distance d0 are said to be gapless.\(\triangle\)

Definition: A nonempty set \(M \subseteq \mathbb{R}^{n}\) for \(n \in \mathbb{N}^{*}\) is said to be \(h\)-homogeneous and is denoted by \(h\)-\(M\) if the minimum distance between any two of its neighbouring points is \(h \in \mathbb{R}_{>0}\) in each dimension. For subsets of \(\mathbb{C}^{n}\), the isomorphism \(\mathbb{C}^{n} \cong \mathbb{R}^{2n}\) causes this analogously. 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. Let \(\dot{\mathbb{K}}^n\) the point-symmetric \(\mathbb{K}^n.\triangle\)

Remark: To \(h\)-homogenise a set, move \(h\) away from the origin in each dimension and round elements in between to the nearest \(h\)-homogeneous elements. Moreover \({}^{\nu}\mathbb{A}_{\mathbb{Q}} \subset {}^{\nu}\mathbb{Q}\) is true and 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 \(\mathbb{R} = \mathbb{Q}\) is a maximal, linearly ordered, closed, continuous and \(d0\)-homogeneous field giving \(|\mathbb{R}| = 2 {\varsigma}^{2} + 1\).

Proof: If \(h\)-homogenised elements have distance d0, continuity is given.\(\square\)

Remark: Therefore, irrational numbers do not exist and \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\) as well as \(\mathbb{C}\) are \(h\)-homogeneous: the Erdős-Ulam problem is solved. Since reconstructing rational 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: Every non-empty set can be linearly ordered.\(\square\)

Remark: Vividly, this is a threading of pearls into a chain. It does not matter how many dimensions the mentioned set has. 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 d0 \(|\mathbb{R}|^n\). Another linear order is the spiral order starting at \((0, …, 0)^T\) according to the Euclidean norm for individual points. The Jacobian conjecture leads to the1cf. Knabner, Peter; Barth, Wolf: Lineare Algebra; 2., überarb. und erw. Aufl.; 2018; Springer; Berlin, p. 380

Theorem: Exactly unitary matrices from \(\mathbb{C}^{n\times n}\) effect d0-continuous bijections on \({\dot{\mathbb{C}}}^n.\square\)

© 2002-2021 by Boris Haase

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References

References
1 cf. Knabner, Peter; Barth, Wolf: Lineare Algebra; 2., überarb. und erw. Aufl.; 2018; Springer; Berlin, p. 380