Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.18511

Bolzano's Conjecture: Measuring the Numerosity of Infinite Sets

Julian Jack
View original source ↗Read source document ↗Export metadata ↓

Abstract

Bolzano and Cantor were the first mathematicians to make significant attempts to measure the size (numerosity) of different infinite collections. They differed in their methodological approaches, with Cantor's prevailing. This led to the foundation of the theory of sets as well as Cantor's transfinite arithmetic. This paper argues that Bolzano's conjecture is correct and that Euclid's principle, 'that the whole is greater than a part', should be considered as a necessary condition for the quantification of infinite sets (rather than bijection). Cantor had concluded that the rational and the algebraic numbers were of the same size as the natural numbers, whilst, in contrast, the real numbers were a larger set. Using Cantor's methods it is shown in this paper that the rational numbers are of larger size than the natural numbers, thus showing that bijection is not a reliable measure of the size of infinite sets. It is also concluded, using mathematical induction, that different 'countably' infinite sets can have various different sizes. The implications for theorems using bijection as a measure of size is then briefly discussed. There already exist new methods of measuring numerosity, based on Euclid's principle, which may develop a consistent system of infinite arithmetic.

Explore related subjects

math.GM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Julian Jack. 2024-03-27. Bolzano's Conjecture: Measuring the Numerosity of Infinite Sets. https://arxiv.org/abs/2403.18511

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗