Bolzano's Conjecture: Measuring the Numerosity of Infinite Sets

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.