Related papers: Bolzano's Infinite Quantities
In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
We present a generalisation of the theory of quantitative algebras of Mardare, Panangaden and Plotkin where (i) the carriers of quantitative algebras are not restricted to be metric spaces and can be arbitrary fuzzy relations or generalised…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…
Boltzmann's equation provides a microscopic model for the evolution of dilute classical gases. A fundamental problem in mathematical physics is to rigorously derive Boltzmann's equation starting from Newton's laws. In the 1970s, Oscar…
It is an old question how massive polynomial hulls of Cantor sets in $\mathbb{C}^n$ can be. In contrast to expectation e.g. Rudin, Vitushkin and Henkin showed on examples that it can be rather massive. Motivated by problems of holomorphic…
We provide a pure algebraic version of the dynamical characterization of Conrad's property. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof…
We introduce a model of simple type theory with potential infinite carrier sets. The functions in this model are automatically continuous, as defined in this paper. This notion of continuity does not rely on topological concepts, including…
After about a century since the first attempts by Bohr, the interpretation of quantum theory is still a field with many open questions. In this article a new interpretation of quantum theory is suggested, motivated by philosophical…
For alternate Cantor real base numeration systems we generalize the result of Frougny and~Solomyak on~arithmetics on the set of numbers with finite expansion. We provide a class of alternate bases which satisfy the so-called finiteness…
Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume's Principle is analytic then in the standard setting the answer appears to be yes. Hodes's work pointed to a way out by…
Cantor's famous construction of the real continuum in terms of Cauchy sequences of rationals proceeds by imposing a suitable equivalence relation. More generally, the completion of a metric space starts from an analogous equivalence…
All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit…
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…
We obtain partial progress towards answering the question of whether the quantity defined in the Mondrian Puzzle can ever equal 0. More specifically, we obtain a nontrivial lower bound for the cardinality of the set $\{n\leq x: M(n)\neq0…
In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with…
Inspired by a question of Kra, Moreira, Richter, and Robertson, we prove two new results about infinite polynomial configurations in large subsets of the rational numbers. First, given a finite coloring of $\mathbb{Q}$, we show that there…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
In his remarkable paper Formalism64, Robinson defends his philsophocal position as follows: (i) Any mention of infinite totalities is literally meaningless. (ii) We should act as if infinite totalities really existed. Being the originator…
By using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in $R^{\infty}$ described in [G.R. Pantsulaia, On uniformly distributed sequences of an increasing family of finite…