Quantum entropy and cardinality of the rational numbers

Abstract

We compare two methods for evaluating the cardinality of the Cartesian product $N \times N$ of the set of natural numbers $N$. The first is used to explain the thermodynamics of black body radiation by using convergent functions on $N \times N$. The cardinality of $N \times N$ enters through the partition function, internal energy and entropy for every macrostate given by a normal mode of electromagnetic wave. Here, $N \times N$ is assigned a greater cardinality than $N$. The second method was devised in analysis to count the rational numbers by using divergent functions on $N \times N$. Here, $N \times N$ is not assigned a greater cardinality than $N$. In this article, we show that the experimentally confirmed first approach is mathematically more consistent with the definition of the real line and foundations of topology. It also provides a quantitative measure of the cardinality of $N \times N$ relative to that of N. Similar arguments show that the set of rational numbers is not countable. This article suggests that the axiom of choice is a more rigorous technique to prove the existence theorems for connection and metric on the spacetime manifold than the usual application of second-countability.