Related papers: The Uncountability of the Unit Interval
Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize…
A rather easy yet rigorous proof of a version of G\"odel's first incompleteness theorem is presented. The version is "each recursively enumerable theory of natural numbers with 0, 1, +, *, =, logical and, logical not, and the universal…
It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i \in [0,q)$ satisfies the equality $\sum_{i=1}^\infty c_iq^{-i}=1$. The set of such "univoque numbers" has…
For any analytically finite Riemann surface, the Teichm\"uller modular group is countable, but it is not easy to find an analytically infinite Riemann surface for which the Teichm\"uller modular group is countable. In this paper, we show…
We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements…
By the example of the proof of Minkowski's conjecture on critical determinant we give a category theory framework for interval computation.
Cantor's first idea to build a one-to-one mapping from the unit interval to the unit square did not work since, as pointed out by Dedekind, the so-obtained function is not surjective. Here, we start from this function and modify it (on a…
In a recent paper [1], it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum…
The three distance theorem states that for any given irrational number $\alpha$ and a natural number $n$, when the interval $( 0, 1 )$ is divided into $n+1$ subintervals by integer multiples of $\alpha$, namely, $\{0\}, \{ \alpha \}, \{…
Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the…
A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into…
The representation of a given quantity with less information is often referred to as `quantization' and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit…
Quantum physics is surprising in many ways. One surprise is the threat to locality implied by Bell's Theorem. Another surprise is the capacity of quantum computation, which poses a threat to the complexity-theoretic Church-Turing thesis. In…
We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with…
The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and…
This paper examines whether unitary evolution alone is sufficient to explain emergence of the classical world from the perspective of computability theory. Specifically, it looks at the problem of how the choice related to the measurement…
We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.
In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case…
Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to…
We consider the fragment F of first order arithmetic in which quantification is restricted to ''for all but finitely many.'' We show that the integers form an F-elementary substructure of the real numbers. Consequently, the F-theory of…