Related papers: Bolzano and the Part-Whole Principle
A `whole-part' theory is developed for a set of finite quantum systems $\Sigma (n)$ with variables in ${\mathbb Z}(n)$. The partial order `subsystem' is defined, by embedding various attributes of the system $\Sigma (m)$ (quantum states,…
Quantification starts with sum and product rules that express combination and partition. These rules rest on elementary symmetries that have wide applicability, which explains why arithmetical adding up and splitting into proportions are…
A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function…
In this paper we prove that the set of countable bqos (viewed as a subset of the Cantor space) is Pi^1_2-complete. The notion of bqo or better quasi-ordering arises from combinatorics and is a generalization of the canonical example of a…
We review and compare five ways of assigning totally ordered sizes to subsets of the natural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and $\alpha$-numerosity. Generalised…
We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which…
We study the quantum measurement problem in the context of an infinite, statistically uniform space, as could be generated by eternal inflation. It has recently been argued that when identical copies of a quantum measurement system exist,…
Wolfram's Principle of Computational Equivalence (PCE) implies that universal complexity abounds in nature. This paper comprises three sections. In the first section we consider the question why there are so many universal phenomena around.…
Bohr's complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental setup, a quantum system (or quanton) can exhibit either its wave-like character, denoted as $W$, or its…
Alternative set theory was created by the Czech mathematician Petr Vop\v enka in 1979 as an alternative to Cantor's set theory. Vop\v enka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set…
A new notion of partition-determined functions is introduced, and several basic inequalities are developed for the entropy of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these…
Over the course of the last 50 years, many questions in the field of computability were left surprisingly unanswered. One example is the question of $P$ vs $NP\cap co-NP$. It could be phrased in loose terms as "If a person has the ability…
A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The developed approach has a pronounced applied character and is based on the principle `The…
A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle `The part is less than the whole' introduced by Ancient Greeks and applied to all…
A mathematical framework is proposed for the "big bang". It starts with some Cantor set and assumes the existence of a transformation from that set to the continuum set used in conventional physical theories like general relativity and…
A quotient of a poset $P$ is a partial order obtained on the equivalence classes of an equivalence relation $\theta$ on $P$; $\theta$ is then called a congruence if it satisfies certain conditions, which vary according to different…
Orbit-finite models of computation generalise the standard models of computation, to allow computation over infinite objects that are finite up to symmetries on atoms, denoted by $\mathbb{A}$. Set theory with atoms is used to reason about…
We investigate quantization coefficients for self-similar probability measures \mu on limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization…
Even a century after the formulation of Quantum Mechanics (QM), the wave function collapse (WFC) remains a contentious aspect of the theory. Environment-induced decoherence has offered a partial resolution by illustrating how unitary…
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…