Related papers: Cantor's Non-Equinumerosity Theorems, Inductively
Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in…
In three papers Colbeck and Renner (Nature Communications 2:411, (2011); Phys. Rev. Lett. 108, 150402 (2012); arXiv:1208.4123) argued that "no alternative theory compatible with quantum theory and satisfying the freedom of choice assumption…
Using an iterative tree construction we show that for simple computable subsets of the Cantor space Hausdorff, constructive and computable dimensions might be incomputable.
In this article we consider alternative definitions-descriptions of a set being Infinite within the primitive Axiomatic System of Zermelo.
We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no…
We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in…
This text tries to give an elementary introduction to the mathematical properties of infinite sets. The aim is to keep the approach as simple as possible. Advanced knowledge of mathematics is not necessary for a proper understanding, and…
We characterize countable dimensionality and strong countable dimensionality by means of an infinite game.
We prove the analogue of Schanuel's conjecture for raising to the power of an exponentially transcendental real number. All but countably many real numbers are exponentially transcendental. We also give a more general result for several…
A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we…
We discuss counting problems linked to finite versions of Cantor's diagonal of infinite tableaux. We extend previous results of [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative…
In the context of a physical theory, two devices, A and B, described by the theory are called incompatible if the theory does not allow the existence of a third device C that would have both A and B as its components. Incompatibility is a…
We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is…
It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…
Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an…
Let $m\geq 3$ be a positive integer. We prove that there are uncountably many non-commensurable metabelian uniform pro-$p$ groups of dimension $m$. Consequently, there are uncountably many non-commensurable finitely presented pro-$p$ groups…
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…
The \emph{International Obfuscated C Code Contest} was a programming contest for the most creatively obfuscated yet succinct C code. By \emph{contrast}, an interest herein is in programs which are, \emph{in a sense}, \emph{easily} seen to…
We prove Euler's theorem of number theory developing an argument based on quandles. A quandle is an algebraic structure whose axioms mimic the three Reidemeister moves of knot theory.
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…