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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.12972 |
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| _version_ | 1866915948950192128 |
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| author | Ghosh, Kaushik |
| author_facet | Ghosh, Kaushik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12972 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum entropy and cardinality of the rational numbers Ghosh, Kaushik General Physics 58A05, 81P17, 26A03, 03E10, 03E25 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. |
| title | Quantum entropy and cardinality of the rational numbers |
| topic | General Physics 58A05, 81P17, 26A03, 03E10, 03E25 |
| url | https://arxiv.org/abs/2509.12972 |