Related papers: Remarks on Cantor's diagonalization proof of 1891
It is well known that permutations avoiding any 3-length pattern are enumerated by the Catalan numbers. If the three patterns 123, 132 and 213 are avoided at the same time we obtain a class of permutations enumerated by the Fibonacci…
The Turing machine (TM) and the Church thesis have formalized the concept of computable number, this allowed to display non-computable numbers. This paper defines the concept of number "approachable" by a TM and shows that some (if not all)…
Real numbers do not admit an extensional procedure for observing discrete information, such as the first digit of its decimal expansion, because every extensional, computable map from the reals to the integers is constant, as is well known.…
The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of…
Representing real numbers using convenient numeration systems (integer bases, $\beta$-numeration, Cantor bases, etc.) has been a longstanding mathematical challenge. This paper focuses on Cantor real bases and, specifically, on automatic…
We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts…
We question the commonly accepted statement that random numbers certified by Bell's theorem carry some special sort of randomness, so to say, quantum randomness or intrinsic randomness. We show that such numbers can be easily generated by…
The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical…
A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. H. Barendregt has shown that if we can represent, for a numeral system, the functions : Successor,…
The ternary Cantor set $C$, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article we will explore the…
In connection with recent work on gaps in the asymptotic subranks of complex tensors the question arose whether the number of nonnegative real numbers that arise as the asymptotic subrank of some complex tensor is countable. In this short…
A sequence of non-negative integers is exactly realizable as the fixed point counts sequence of a dynamical system if and only if it gives rise to a sequence of non-negative orbit counts. This provides a simple realizability criterion based…
We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen…
We show that the real line R viewed as a vector space is of uncountable (algebraic) dimension over the scalar field Q of rational numbers. We then build an operator J which maps {R, Q} onto {R, Q}, is Q-linear and whose graph is scattered…
In problem 10621 of the American Mathematical Monthly the question was raised whether it is possible to explicitly compute Cantor's singular moment of order -1. This is answered in the affirmative.
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
The impossibility of eliminating hallucination, understood here as incorrect definite answers, in sufficiently expressive yes-or-no formal domains is an immediate consequence of classical undecidability theorems. This note does not revisit…
We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over $\mathbb{F}_q$ and their classification. Through a mix of linear programming,…
In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for…
Imagine a sequence in which the first letter comes from a binary alphabet, the second letter can be chosen on an alphabet with 10 elements, the third letter can be chosen on an alphabet with 3 elements and so on. When such a sequence can be…