Drafts by Wolfgang Mückenheim
It is shown that not all numbers can be expressed and communicated such that the receiver knows w... more It is shown that not all numbers can be expressed and communicated such that the receiver knows what the sender has meant. We call them dark numbers.
Evidence of dark numbers, 2024
This work contains several arguments for the existence of dark numbers, i.e., numbers which canno... more This work contains several arguments for the existence of dark numbers, i.e., numbers which cannot be manipulated as individuals but only collectively. Their existence depends on the premise of actual infinity. Whether actually infinite sets exist is unknown and cannot be proven; it can only be assumed as an axiom. But if actually infinite sets exist then dark elements are unavoidable. This concept helps to explain many paradoxes of set theory like Zeno's paradox or the paradox of the binary tree or the "completely scattered space" of the real axis.
We will prove by means of Cantor's mapping between natural numbers and positive fractions that hi... more We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.
Cantor's idea of actual infinity turns the potentially infinite sequence of natural numbers into ... more Cantor's idea of actual infinity turns the potentially infinite sequence of natural numbers into an actually infinite set. But this implies that almost all natural numbers cannot be instantiated, expressed by digits or sums of units, and connected by finite initial segments without any interruption to the origin 0.
Papers by Wolfgang Mückenheim
viXra, Mar 1, 2017
Limits of sequences of sets required to define infinite bijections do not only raise paradoxes bu... more Limits of sequences of sets required to define infinite bijections do not only raise paradoxes but cause self-contradictory results.
Transfinity - A Source Book, 2023
Transfinity is the realm of numbers larger than every natural number: For every natural number k ... more Transfinity is the realm of numbers larger than every natural number: For every natural number k there are infinitely many natural numbers n > k. For a transfinite number t there is no natural number n t.
We will first present the theory of actual infinity, mainly sustained by quotes, in chapter I and then transfinite set theory as far as necessary to understand the following chapters. In addition the attitude of the founder of transfinite set theory, Georg Cantor, with respect to sciences and religion (his point of departure) will be illuminated by various quotes of his as well as of his followers in chapter IV. Also the set of applications of set theory will be summarized there. All this is a prerequisite to judge the social and scientific environment and the importance of set theory. Quotes expressing a sceptical attitude against transfinity or addressing questionable points of current mathematics based on it are collected in chapter V. For a brief overview see also Critics of transfinity. The critique is scrutinized in chapter VI, the main part of this source book. It contains over 100 arguments against actual infinity – from doubtful aspects to clear contra¬dictions – among others applying the newly devised powerful method of ArithmoGeometry. Finally we will present in chapter VII MatheRealism, a theory that shows that in real mathematics, consisting of monologue, dialogue, and discourse between real thinking-devices, via necessarily physical means, infinite sets cannot exist other than as names. This recognition removes transfinity together with all its problems from mathematics – although the application of mathematics based on MatheRealism would raise a lot of technical problems.
viXra, Feb 1, 2017
It is shown that the enumeration of rational numbers cannot be complete. The impossibility of so-... more It is shown that the enumeration of rational numbers cannot be complete. The impossibility of so-called supertasks [1] is generally accepted. By the equivalence of spatial and temporal axes it is clear already that bijections between different infinite sets are impossible too. But here we will give an independent proof. Cantor's enumeration of the set-+ of positive rational numbers q is ordered by the ascending sum (a+b) of numerator a and denominator b of q = a/b, and in case of equal sum, by ascending numerator a. Since all fractions will repeat themselves infinitely often, repetitions will be dropped when enumerating the rational numbers.
viXra, Mar 1, 2017
Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of s... more Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ. We will show this for the set Ù of natural numbers by the newly devised powerful tool of arithmogeometry as well as by scrutinizing the recursive construction. The basic theorem of set theory "n oe Ù: n < ¡ 0 precludes the defined identity |Ù| = ¡ 0. Further differences of limits in set theory and analysis are discussed.
We will prove by means of Cantor's mapping between natural numbers and positive fractions that hi... more We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.
Über Mathematik und Wirklichkeit und dieses Buch
Mathematik für die ersten Semester, 2015
Mathematik für die ersten Semester
Some Arguments against the Existence of de Broglie Waves
Wave-Particle Duality, 1992
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Drafts by Wolfgang Mückenheim
Papers by Wolfgang Mückenheim
We will first present the theory of actual infinity, mainly sustained by quotes, in chapter I and then transfinite set theory as far as necessary to understand the following chapters. In addition the attitude of the founder of transfinite set theory, Georg Cantor, with respect to sciences and religion (his point of departure) will be illuminated by various quotes of his as well as of his followers in chapter IV. Also the set of applications of set theory will be summarized there. All this is a prerequisite to judge the social and scientific environment and the importance of set theory. Quotes expressing a sceptical attitude against transfinity or addressing questionable points of current mathematics based on it are collected in chapter V. For a brief overview see also Critics of transfinity. The critique is scrutinized in chapter VI, the main part of this source book. It contains over 100 arguments against actual infinity – from doubtful aspects to clear contra¬dictions – among others applying the newly devised powerful method of ArithmoGeometry. Finally we will present in chapter VII MatheRealism, a theory that shows that in real mathematics, consisting of monologue, dialogue, and discourse between real thinking-devices, via necessarily physical means, infinite sets cannot exist other than as names. This recognition removes transfinity together with all its problems from mathematics – although the application of mathematics based on MatheRealism would raise a lot of technical problems.