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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.\(\triangle\)

Remark: Let be an open question whether there are actual infinite sets or only potentially infinite ones, since a proof is not possible because of the transcendence of the infinite. An abrupt transition between finite and infinite numbers is hard to justify. 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 \(\mathbb{R}\), which includes both the conventionally hyperreal and surreal numbers, 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 \(\mathbb{Q} + i\mathbb{Q}\). Between the finite and the infinite numbers, there are the inconcrete ones. For reasons of clarity, the underlying set will be specified after an interval.\(\triangle\)

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 \(c := 2^\hslash\) be for \(\hslash \in \mathbb{N}^{*}\) the largest finite, \(\omega := 2^\ell\) for \(\ell \in \mathbb{N}^{*}\) the largest 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. Let d0 \(:= 2^{-\wp} = \min \mathbb{R}_{>0} \) the 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 \(x_0 := 2.\) Let \({}^{\omega}\mathbb{R} := [-\omega, \omega]\mathbb{R}\) and \({}^{\omega}\mathbb{C} := {}^{\omega}\mathbb{R} + i{}^{\omega}\mathbb{R} \subset \mathbb{C}\). If \({}^{\omega}\) precedes a real or complex set in the following text, it denotes the intersection with \([-\omega, \omega]\mathbb{R}\) or \({}^{\omega}\mathbb{C}\).\(\triangle\)

Remark: Minimality theorem (see number theory) and digital computer explain the choice of 2. The latter sets only contain non-infinite elements. The notation with \({}^{c}\) is defined analogously for \([-c, c]\mathbb{R}\) or \({}^{c}\mathbb{C}\). 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. The almost arbitrary definitions of the largest finite and (non-)infinite real number have no convincing alternatives.

Remark: The number of elements of an inconcrete or infinite set can be only uniquely determined if its construction is known. Then the set can be compared to \({}^{\omega}\mathbb{N}\), which may be taken as a basis thanks to its simple construction. If there are multiple possible constructions, the most plausible one wins the bid. The set of all sets cannot be changed.

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. Sets of dual numbers begin with \(d\).\(\triangle\)

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).\(\triangle\)

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 \({a}_{k} \in {}^{c}\mathbb{Z}\) 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.\(\triangle\)

Definition: The numbers \(z \in \mathbb{C}\) setting \(p(z)\) are called zeros and 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 one. The special case \({a}_{\deg(p)} = 1\) yields \(m\)-algebraic integers. 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.\(\triangle\)

Definition: The notation for \(m\)-algebraic numbers is \({(m, {a}_{k-1}, {a}_{k-2}, ..., {a}_{1}, {a}_{0}; r, i; \#n, \&q; v, p)}_{s}\), 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. Here \(r\) takes 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, \(\&q\) gives the number \(q \in {}^{c}\mathbb{N}\) of repeated zeros. For \(k\)-minimal polynomials or series, the sign < is taken as the value of the specification \(s\), in every other case, the sign > is used. The numerical value \(v\) has the precision \(p\).\(\triangle\)

Remark: Not distinguishing between repeated zeros allows the zeros of \(k\)-polynomials or \(k\)-series with integer or rational coefficients to be endowed with a strict total ordering. The information \(r, i, \#n, \&q, v, p\), and \(s\) can optionally be omitted as e.g. for rational numbers. The \((c+2)\)-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: Its relative complement shows the elements with missing partner element for the bijection.\(\square\)

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. Because there are infinitely many sets whose number of elements lie between \(|{}^{c}\mathbb{N}|\) and \(|{}^{c}\mathbb{R}|\), the continuum hypothesis is wrong, too.

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: 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 area or half of the circumference of the unit circle defines pi \(\pi\). Let \(\iota := \pi/2\). Euler's number \(e\) as the solution of the equation \({x}^{i\pi} = -1\). Then the logarithm function \(\ln\) is defined 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.\(\triangle\)

Remark: In calculations, typically resorting to approximations will be necessary. The preceding definition is \(\mathcal{O}(\hat{c})\) larger than the also possible one where \(e := {(1 + \hat{c})}^{c}\). The exponential series being exactly differentiated with as many terms as possible justifies the former. This deviation can have negative consequences when attempting to calculate as precisely as possible. Adjoining the symbol \(\infty \gg \varsigma^2\) to the real numbers allows calculations like having a constant. If \(\pm0\) is replaced by \(\pm\hat{\infty}\), the calculations become unique and consistent.

Remark: This avoids a division by 0 and any vague notions of limits, but requires the careful consideration where this replacement makes sense, and not switching arbitrarily between symbols. This also allows to define integrals and differentials for each operation on real and complex numbers in such a way that every function is at least directionally integrable and differentiable wherever the function values are defined (see Nonstandard Analysis).

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 \({a}_{m}{x}^{m} + {a}_{\acute{m}}{x}^{\acute{m}} + ... + {a}_{1}x + {a}_{0}\) 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 for the Riemann zeta function \(\zeta\) 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 \(z(m)\) 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 \(p\), the number of polynomials or series must be multiplied 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 \(m = 1\), there are \(3(\hat{\iota}n)^{2}+\mathcal{O}(n \text{ ln }n)\) rational solutions. For \(m = 2\), \({\frac{9}{2}n}^{3}/\zeta(3) + \mathcal{O}({n}^{2}\ln \, n)\) real solutions arise, since a real polynomial of degree 2 has two real zeros with probability \({\frac{9}{16}}\) by the quadratic formula. For \({a}_{m} = 1\), there are \(z(m){(2n+1)}^{\acute{m}}(2n + \mathcal{O}(\ln \, n))\) 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; Bull. Amer. Math. Soc. 49 (4); 1943; 314 - 320).

Examples: For \(m = n = \acute{c} =: e^{\mathrm{\nu}} =: \acute{\kappa}/2\), it is true that \(|{}^{c}\mathbb{A}_{\mathbb{R}}| = \frac{\mathrm{\nu}}{\mathrm{\iota}}{\mathrm{\kappa}^{\acute{c}}}\left(c+\mathcal{O}(\mathrm{\nu})\right)\) and \(|{}^{c}\mathbb{A}_{\mathbb{C}}| = {\frac{1}{2}} {\kappa}^c\left(c+\mathcal{O}(\mathrm{\nu})\right)\).

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 \({}^{c}\mathbb{A}_{\mathbb{Q}} \subset {}^{c}\mathbb{Q}\) 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, 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 \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}\) and \(\mathbb{C}\) are \(h\)-homogeneous. This refutes Fermat's conjecture.

© 2002-2018 by Boris Haase


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