On Accuracy and Coherence with Infinite Opinion Sets

There is a well-known equivalence between avoiding accuracy dominance and having probabilistically coherent credences (see, e.g., de Finetti 1974, Joyce 2009, Predd et al. 2009, Pettigrew 2016). However, this equivalence has been established only when the set of propositions on which credence functions are defined is finite. In this paper, I establish connections between accuracy dominance and coherence when credence functions are defined on an infinite set of propositions. In particular, I establish the necessary results to extend the classic accuracy argument for probabilism to certain classes of infinite sets of propositions including countably infinite partitions.

A new look at infinitely large and infinitely small quantities: Methodological foundations and practical calculations with these numbers on a computer

This article describes a recently proposed methodology that allows one to work with infinitely large and infinitely small quantities on a computer. The approach uses a number of ideas that bring it closer to modern physics, in particular, the relativity of mathematical knowledge and its dependence on the tools used by mathematicians in their studies are discussed. It is shown that the emergence of new computational tools influences the way we perceive traditional mathematical objects, and also helps to discover new interesting objects and problems. It is discussed that many difficulties and paradoxes regarding infinity do not depend on its nature, but are the result of the weakness of the traditional numeral systems used to work with infinitely large and infinitely small quantities. A numeral system is proposed that not only allows one to work with these quantities analytically in a simpler and more intuitive way, but also makes possible practical calculations on the Infinity Computer, patented in a number of countries. Examples of measuring infinite sets with the accuracy of one element are given and it is shown that the new methodology avoids the appearance of some well-known paradoxes associated with infinity. Examples of solving a number of computational problems are given and some results of teaching the described methodology in Italy and Great Britain are discussed.

Why is Generative Grammar Recursive?

A familiar argument goes as follows: natural languages have infinitely many sentences, finite representation of infinite sets requires recursion; therefore any adequate account of linguistic competence will require some kind of recursive device. The first part of this paper argues that this argument is not convincing. The second part argues that it was not the original reason recursive devices were introduced into generative linguistics. The real basis for the use of recursive devices stems from a deeper philosophical concern; a grammar that functions merely as a metalanguage would not be explanatorily adequate as it would merely push the problem of explaining linguistic competence back to another level. The paper traces this concern from Zellig Harris and Chomsky’s early work in generative linguistics and presents some implications.

Finitely Supported Binary Relations between Infinite Atomic Sets

In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the T-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations between infinite atomic sets. Of particular interest are finitely supported Dedekind-finite sets because they do not contain finitely supported, countably infinite subsets. We prove that the infinite sets ℘fs(Ak×Al), ℘fs(Ak×℘m(A)), ℘fs(℘n(A)×Ak) and ℘fs(℘n(A)×℘m(A)) do not contain uniformly supported infinite subsets. Moreover, the functions space ZAm does not contain a uniformly supported infinite subset whenever Z does not contain a uniformly supported infinite subset. All these sets are Dedekind-finite in the framework of finitely supported structures.

The way to list all infinite real numbers and to construct a bijection between natural numbers and real numbers

Georg Cantor defined countable and uncountable sets for infinite sets. The set of natural numbers is defined as a countable set, and the set of real numbers is proved to be uncountable by Cantor’s diagonal argument. Most scholars accept that it is impossible to construct a bijection between the set of natural numbers and the set of real numbers. However, the way to construct a bijection between the set of natural numbers and the set of real numbers is proposed in this paper. The set of real numbers can be proved to be countable by Cantor’s definition. Cantor’s diagonal argument is challenged because it also can prove the set of natural numbers to be uncountable. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

A new perspective of countable and uncountable infinite sets on Georg Cantor’s definition in set theory

Georg Cantor defined countable and uncountable sets for infinite sets. Natural number set is defined as a countable set, and real number set is proven as an uncountable set by Cantor’s diagonal method. However, in this paper, natural number set will be proven as an uncountable set using Cantor’s diagonal method, and real number set will be proven as a countable set by Cantor’s definition. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

A new perspective of countable and uncountable infinite set on Georg Cantor’s definition in set theory

Georg Cantor defined countable and uncountable sets for infinite sets. Natural number set is defined as a countable set, and real number set is proven as an uncountable set by Cantor’s diagonal method. However, in this paper, natural number set will be proven as an uncountable set using Cantor’s diagonal method, and real number set will be proven as a countable set by Cantor’s definition. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

Integral geometry on discrete matrices

In this note, we study the Radon transform and its dual on the discrete matrices by defining hyperplanes as being infinite sets of solutions of linear Diophantine equations. We then give an inversion formula and a support theorem.