Related papers: The Uncountability of the Unit Interval
In his first set theory paper (1874), Cantor establishes the uncountability of $\mathbb{R}$. We study the latter in Kohlenbach's higher-order Reverse Mathematics, motivated by the observation that one cannot study concepts like `arbitrary…
We show that there exist real quadratic maps of the interval whose attractors are computationally intractable. This is the first known class of such natural examples.
We investigate the minimum cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On binary case, we follow the…
This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…
In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…
The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one…
We construct a topos in which the Dedekind reals are countable. The topos arises from a new kind of realizability, which we call parameterized realizability, based on partial combinatory algebras whose application depends on a parameter.…
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…
Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is…
We give an uncountability proof of the reals which relies on their order completeness instead of their sequential completeness. We use neither a form of the axiom of choice nor the law of excluded middle, therefore the proof applies to the…
A physical system is determined by a finite set of initial conditions and "laws" represented by equations. The system is computable if we can solve the equations in all instances using a "finite body of mathematical knowledge". In this…
In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.
We study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all…
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…
Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. Upon examining these examples in the context of infinities from Cantor's theory of transfinite numbers, the only known…
This paper provides some counterexamples to Cantor's contributions to the foundations of Set Theory. The first counterexample forces Cantor's Diagonal Method (DM) to yield one of the numbers in the target list. To study this anomaly, and…
The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the…
This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal…
This article could be called "theme and variations" on Cantor's celebrated diagonal argument. Given a square nxn tableau T=(a_i^j) on a finite alphabet A, let L be the set of its row-words. The permanent Perm(T) is the set of words…
Generalizing a geometric idea due to J. Sondow, we give a geometric proof for the Cantor's Theorem. Moreover, it is given an irrationality measure for some Cantor series.