The Infinite in Sciences and Arts
Actual infinity in its various forms is discussed, searched and not found.
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arXiv subjects
W. Mueckenheim
Publications and source records attributed to W. Mueckenheim.
Infinite sets are non-denumerable
Cantor's famous proof of the non-denumerability of real numbers does apply to any infinite set. The set of exclusively all natural numbers does not exist. This shows that the concept of countability is not well defined. There remains no evidence for the existence of transfinite cardinal numbers.
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On Cantor's important proofs
It is shown that the pillars of transfinite set theory, namely the uncountability proofs, do not hold. (1) Cantor's first proof of the uncountability of the set of all real numbers does not apply to the set of irrational numbers alone, and, therefore, as it stands, supplies no distinction between the uncountable set of irrational numbers and the countable set of rational numbers. (2) As Cantor's second uncount-ability proof, his famous second diagonalization method, is an impossibility proof, a simple counter-example suffices to prove its failure. (3) The contradiction of any bijection between a set and its power set is a consequence of the impredicative definition involved. (4) In an appendix it is shown, by a less important proof of Cantor, how transfinite set theory can veil simple structures.
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The Meaning of Infinity
The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural numbers which it is ascribed to, cannot contain an actually infinite number of elements. Further the basic inequality of transfinite set theory aleph0 < 2^aleph0 is found invalid, and, consequently, the set of real numbers is proven denumerable by enumerating it.
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Physical Constraints of Numbers
All sciences need and many arts apply mathematics whereas mathematics seems to be independent of all of them, but only based upon logic. This conservative concept, however, needs to be revised because, contrary to Platonic idealism (frequently called "realism" by mathematicians), mathematical ideas, notions, and, in particular, numbers are not at all independent of physical laws and prerequisites.
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A severe inconsistency of transfinite set theory
Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well ordered. (3) In a finite set of real numbers the maximum below a given limit can always be determined. (4) Any two different real numbers are separated by at least one rational number. These theorems are applied to map the irrational numbers into the rational numbers, showing that the set of all irrational numbers is countable.
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On Cantor's Theorem
The famous contradiction of a bijection between a set and its power set is a consequence of the impredicative definition involved. This is shown by the fact that a simple mapping between equivalent sets does also fail to satisfy the critical requirement.
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On the abundance of the irrational numbers
Based upon the axiom of choice it is proved that the cardinality of the rational numbers is not less than the cardinality of the irrational numbers. This contradicts a main result of transfinite set theory and shows that the axiom of choice is invalid.
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