Quantum entropy and cardinality of the rational numbers

Kaushik Ghosh

arxiv: 2509.12972 · v5 · submitted 2025-09-16 · ⚛️ physics.gen-ph

Quantum entropy and cardinality of the rational numbers

Kaushik Ghosh This is my paper

Pith reviewed 2026-05-18 16:45 UTC · model grok-4.3

classification ⚛️ physics.gen-ph

keywords cardinalityrational numbersblackbody radiationquantum entropypartition functionaxiom of choicespacetime manifold

0 comments

The pith

Blackbody radiation formulas assign higher cardinality to N times N than to N, making the rationals uncountable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper compares two ways of handling the cardinality of the Cartesian product of natural numbers with itself. One comes from the thermodynamics of blackbody radiation, where the partition function, internal energy, and entropy for each normal mode treat N times N as larger than N through convergent functions. The other comes from standard analysis, where divergent functions are used to enumerate the rationals and assign N times N the same cardinality as N. The author argues that the first approach aligns better with the definition of the real line and the foundations of topology, supplies a quantitative comparison of the two cardinalities, and therefore shows that the rationals cannot be put into one-to-one correspondence with the naturals. The same reasoning favors the axiom of choice over second-countability when proving the existence of a connection and metric on a spacetime manifold.

Core claim

The experimentally confirmed partition function and entropy expressions for blackbody radiation, which incorporate N times N with a cardinality strictly greater than that of N, are mathematically more consistent with the real line and topology than the enumeration that makes the rationals countable, and therefore imply that the rationals are uncountable while also indicating that the axiom of choice supplies a more rigorous basis for existence theorems on spacetime manifolds than second-countability.

What carries the argument

The partition function and entropy formulas for normal modes of electromagnetic waves, which embed N times N and assign it greater cardinality via convergent series.

If this is right

The set of rational numbers cannot be placed in bijection with the set of natural numbers.
There exists a quantitative measure of how much larger the cardinality of N times N is than that of N.
The axiom of choice is a more rigorous tool than second-countability for establishing the existence of a connection and a metric on a spacetime manifold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same preference for physical consistency over formal enumeration could be applied to other infinite sets that appear in physical calculations.
Reconsidering countability assumptions might alter how limits and series are treated in quantum field theory on curved backgrounds.

Load-bearing premise

The appearance of N times N inside the partition function and entropy expressions supplies a valid assignment of cardinality that overrides the usual bijection between N and N times N.

What would settle it

A calculation showing that the blackbody entropy and energy formulas remain unchanged when N times N is replaced by a bijection with N would directly contradict the claim that the thermodynamic treatment assigns higher cardinality.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims blackbody radiation shows the rationals are uncountable via double sums, but this analogy does not engage or refute the standard bijections.

read the letter

The main takeaway is that the author uses the convergent double sums in blackbody partition functions and entropy to assign higher cardinality to N times N than to N, then extends this to conclude the rationals are uncountable. The paper also floats the idea that the axiom of choice would be cleaner than second-countability for proving connections and metrics on spacetime manifolds. That is the central move on offer from the abstract and claims provided. The comparison between the physics treatment of modes and the usual enumeration of rationals is laid out clearly enough to follow the intended contrast. The experimental success of blackbody formulas is noted as supporting evidence for the physics side. Beyond that, there is little new technique or derivation. The soft spot is the core inference itself. Cardinality is settled by the existence of bijections, such as the Cantor pairing function for N times N or the diagonal listing for the rationals. The paper does not identify any place where those constructions break or supply a replacement definition of cardinality that remains consistent with the construction of the reals. Convergence or divergence of particular series is independent of whether a bijection exists. Treating the appearance of N times N in a partition function as a quantitative measure of cardinality therefore rests on an unbridged analogy rather than a direct argument. No internal contradiction in the physics formulas is shown, and the circularity concern stands: the measure is defined by the success of the thermodynamic expressions. This is not a load-bearing formal result or a machine-checked claim, so the central assertion does not hold up on its own terms. The work might interest someone already exploring whether physics can dictate set-theoretic foundations, but it does not supply enough engagement with the existing proofs to move the discussion forward. I would not bring it to a reading group or cite it. It does not look ready for serious refereeing.

Referee Report

4 major / 1 minor

Summary. The paper compares two methods for assigning cardinality to N × N: the thermodynamics of blackbody radiation, which uses convergent sums over modes indexed by N × N in the partition function, internal energy, and entropy and thereby assigns N × N greater cardinality than N, versus the standard analytic enumeration of rationals via divergent functions on N × N that does not. It argues that the experimentally confirmed physical approach is mathematically more consistent with the definition of the real line and foundations of topology, concludes that the rationals are therefore uncountable, and suggests that the axiom of choice provides a more rigorous basis than second-countability for existence theorems on spacetime manifolds.

Significance. If the central claim held, the work would be significant for offering a physically motivated, quantitative criterion for cardinality that could inform foundations of quantum field theory and differential geometry on manifolds. It would also supply an alternative route to existence results that avoids reliance on second-countability.

major comments (4)

[Abstract] Abstract: the assertion that blackbody thermodynamics assigns N × N greater cardinality than N because its partition function and entropy involve convergent sums is not accompanied by any derivation showing where the standard bijection (Cantor pairing function) between N and N × N fails or by an alternative definition of cardinality consistent with ZFC.
[Abstract] Abstract, paragraph contrasting convergent vs. divergent functions: the inference that experimental confirmation of blackbody formulas implies mathematical consistency with the real line and thereby overrides the diagonal enumeration of the rationals does not identify any inconsistency in the usual construction of Q or supply a test that would falsify the standard countability result.
[Abstract] Abstract: the claim that similar arguments show the rationals are uncountable rests on the same unbridged analogy between physical indexing and set cardinality without engaging the explicit bijection between N and Q or the construction of the reals via Dedekind cuts or Cauchy sequences.
[Abstract] Abstract: the suggestion that the axiom of choice is more rigorous than second-countability for proving existence of connection and metric on spacetime manifolds does not demonstrate how the proposed cardinality re-assignment alters the standard topological arguments or resolves any specific existence gap.

minor comments (1)

[Abstract] Notation for the set of natural numbers is not defined at first use; it is unclear whether N includes 0.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We appreciate the referee's detailed feedback on our manuscript. We address each major comment below, offering clarifications based on the physical motivation of our approach and indicating where revisions will strengthen the presentation.

read point-by-point responses

Referee: [Abstract] the assertion that blackbody thermodynamics assigns N × N greater cardinality than N because its partition function and entropy involve convergent sums is not accompanied by any derivation showing where the standard bijection (Cantor pairing function) between N and N × N fails or by an alternative definition of cardinality consistent with ZFC.

Authors: The manuscript contrasts the physical method, where convergent sums over N × N modes are required for thermodynamic quantities to match experiment, with the analytic enumeration. While the Cantor pairing function provides a bijection in ZFC, it does not preserve the convergence essential to blackbody formulas. Our alternative notion of cardinality is defined by the minimal size permitting convergent physical sums. We will add a derivation in the revised manuscript illustrating the incompatibility of the pairing function with observed spectra. revision: yes
Referee: [Abstract] the inference that experimental confirmation of blackbody formulas implies mathematical consistency with the real line and thereby overrides the diagonal enumeration of the rationals does not identify any inconsistency in the usual construction of Q or supply a test that would falsify the standard countability result.

Authors: Experimental confirmation of Planck's law validates convergent double sums over N × N. This aligns with real-valued observables. The diagonal enumeration employs divergent functions incompatible with physical entropy calculations. The inconsistency lies in this mismatch between divergent analytic mappings and convergent physical ones. A falsification test is whether standard countability reproduces thermodynamic limits without adjustments. We will elaborate on this test in the revision. revision: partial
Referee: [Abstract] the claim that similar arguments show the rationals are uncountable rests on the same unbridged analogy between physical indexing and set cardinality without engaging the explicit bijection between N and Q or the construction of the reals via Dedekind cuts or Cauchy sequences.

Authors: The analogy follows because the enumeration of Q uses N × N indexing analogous to radiation modes. Physical convergence assigns higher cardinality to this indexing, implying the same for Q. The manuscript focuses on the physical criterion rather than full engagement with Dedekind cuts or Cauchy sequences. We will add a brief paragraph in the revision addressing these constructions and the distinction from the physical approach. revision: yes
Referee: [Abstract] the suggestion that the axiom of choice is more rigorous than second-countability for proving existence of connection and metric on spacetime manifolds does not demonstrate how the proposed cardinality re-assignment alters the standard topological arguments or resolves any specific existence gap.

Authors: Reassigning Q as uncountable removes the countable basis assumption of second-countability, requiring the axiom of choice to select structures for metrics and connections. This alters standard arguments by eliminating reliance on countable dense sets. The manuscript suggests this direction without detailing every proof alteration. We will expand the discussion section to outline the impact on topological existence results. revision: partial

Circularity Check

2 steps flagged

Blackbody partition-function indexing treated as cardinality assignment that overrides bijections

specific steps

self definitional [abstract]
"Here, N × N is assigned a greater cardinality than N. ... It also provides a quantitative measure of the cardinality of N × N relative to that of N. Similar arguments show that the set of rational numbers is not countable."

The quantitative measure is obtained by inserting the blackbody partition function, internal energy and entropy (which are written as sums over N×N) into the definition of cardinality. The greater cardinality is therefore true by construction once the physical formulas are accepted as the arbiter; the same construction is then applied to the rationals.
fitted input called prediction [abstract]
"The first is used to explain the thermodynamics of black body radiation by using convergent functions on N × N. The cardinality of N × N enters through the partition function, internal energy and entropy for every macrostate given by a normal mode of electromagnetic wave."

The paper fits the thermodynamic expressions to data using sums indexed by N×N, then presents the resulting 'greater cardinality' as a derived result rather than as a direct consequence of the indexing choice already made in the input formulas.

full rationale

The paper's central move equates the appearance of double sums in experimentally successful blackbody formulas with a 'quantitative measure' of cardinality for N×N that is larger than for N. This measure is then used to declare the standard diagonal enumeration invalid and to conclude that Q is uncountable. Because the assignment of cardinality is defined directly by which functions (convergent vs divergent) are chosen in the physical derivation, the conclusion reduces to the input assumption that thermodynamic indexing is the correct arbiter of set size. No independent redefinition of cardinality or demonstration that the Cantor pairing fails is supplied.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper introduces no explicit free parameters but relies on the unstated axiom that thermodynamic partition functions provide a privileged definition of cardinality. It postulates a 'quantitative measure' of relative cardinality without independent evidence outside the blackbody context.

axioms (2)

domain assumption Thermodynamic derivations using convergent functions on N×N correctly determine the cardinality of N×N relative to N.
Invoked in the comparison of the two methods for evaluating cardinality (abstract).
ad hoc to paper Experimental confirmation of blackbody radiation implies mathematical consistency with the definition of the real line.
Used to elevate the physics method over the analysis method.

invented entities (1)

Quantitative measure of the cardinality of N×N relative to N no independent evidence
purpose: To assign a greater cardinality to N×N than to N based on the blackbody partition function
Introduced to reconcile the thermodynamics approach with topology; no falsifiable handle outside the paper is provided.

pith-pipeline@v0.9.0 · 5730 in / 1626 out tokens · 43596 ms · 2026-05-18T16:45:19.029246+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

Similar arguments show that the set of rational numbers is not countable... 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.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

Here, N×N is assigned a greater cardinality than N... Eqs.(7,8,10,13) demonstrate an experimentally observed counter example to the claim that |N×N|<|N|.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

K. Ghosh,Entropy and Cardinality of the Rational Numbers, Talk given at the 2023International Conference on Topology and its Applications, July 3-7, 2023, Nafpaktos, Greece; https://hal.science/hal-05242734

work page 2023
[2]

Ghosh, J

K. Ghosh, J. Phys.: Conf. Ser. 2090, 012037 (2021), https://hal.archives-ouvertes.fr/hal-03092015

work page 2090
[3]

Huang,Statistical Mechanics, Wiley-India Ltd., New Delhi, 2003

K. Huang,Statistical Mechanics, Wiley-India Ltd., New Delhi, 2003

work page 2003
[4]

Jacobson,Basic Algebra I, Dover Publications, Inc., New York, 1985

N. Jacobson,Basic Algebra I, Dover Publications, Inc., New York, 1985

work page 1985
[5]

Munkres,Topology A First Course, Prentice-Hall of India Private Limited, 1994

James R. Munkres,Topology A First Course, Prentice-Hall of India Private Limited, 1994. 9

work page 1994
[6]

Cohen,Set Theory and the Continuum Hypothesis, Dover Publications, Inc., 1994

Paul J. Cohen,Set Theory and the Continuum Hypothesis, Dover Publications, Inc., 1994

work page 1994
[7]

Hocking and Gail S

John G. Hocking and Gail S. Young; Topology (Dover Publications, Inc., New York, 1961)

work page 1961
[8]

Kobayashi and K

S. Kobayashi and K. Nomizu,Foundations of Differential Geometry:V ol.I, Wiley Classics Library, 1991

work page 1991
[9]

Courant and F

R. Courant and F. John,Introduction to Calculus and Analysis:V ol.1, Springer-Verlag New York Inc., 1989

work page 1989
[10]

Ghosh, International Journal of Pure and Applied Mathematics,76, No.2, pp.251 - 260, (2012)

K. Ghosh, International Journal of Pure and Applied Mathematics,76, No.2, pp.251 - 260, (2012)

work page 2012
[11]

Itzykson and J

C. Itzykson and J. B. Zuber,Quantum Field Theory, Dover Publications, Inc. Mineola, 2005

work page 2005
[12]

Tolman,The principals of Statistical Mechanics, Dover Publications, Inc., New York, 1979

Richard C. Tolman,The principals of Statistical Mechanics, Dover Publications, Inc., New York, 1979

work page 1979
[13]

Apostol,Mathematical Analysis, Narosa Publishing House, 1992

Tom M. Apostol,Mathematical Analysis, Narosa Publishing House, 1992

work page 1992
[14]

C. W. Misner, K. S. Thorne and J. A. Wheeler,Gravitation, W.H. Freeman and company, New York, 1970

work page 1970
[15]

S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time, Cambridge University Press, 1973

work page 1973
[16]

R. M. Wald,General Relativity, The University of Chicago Press, Chicago and London, 1984

work page 1984
[17]

Lovelock and H

D. Lovelock and H. Rund,Tensors, Differential Forms, and Variational Principals, Dover Publications, Inc., New York, 1989

work page 1989
[18]

Ghosh, Quantum Stud.: Math

K. Ghosh, Quantum Stud.: Math. Found.11, 625–642 (2024). https://doi.org/10.1007/s40509-024-00340-9

work page doi:10.1007/s40509-024-00340-9 2024
[19]

Quigg,Gauge Theories of the Strong, Weak and Electromagnetic Interactions, Princeton University Press, Princeton and Oxford, 2013

C. Quigg,Gauge Theories of the Strong, Weak and Electromagnetic Interactions, Princeton University Press, Princeton and Oxford, 2013

work page 2013
[20]

Aharonov and D

Y . Aharonov and D. Bohm,Significance of Electromagnetic Potentials in the Quantum Theory, The Physical Review. 115, No. 3, 1959, pp. 485-491

work page 1959

[1] [1]

K. Ghosh,Entropy and Cardinality of the Rational Numbers, Talk given at the 2023International Conference on Topology and its Applications, July 3-7, 2023, Nafpaktos, Greece; https://hal.science/hal-05242734

work page 2023

[2] [2]

Ghosh, J

K. Ghosh, J. Phys.: Conf. Ser. 2090, 012037 (2021), https://hal.archives-ouvertes.fr/hal-03092015

work page 2090

[3] [3]

Huang,Statistical Mechanics, Wiley-India Ltd., New Delhi, 2003

K. Huang,Statistical Mechanics, Wiley-India Ltd., New Delhi, 2003

work page 2003

[4] [4]

Jacobson,Basic Algebra I, Dover Publications, Inc., New York, 1985

N. Jacobson,Basic Algebra I, Dover Publications, Inc., New York, 1985

work page 1985

[5] [5]

Munkres,Topology A First Course, Prentice-Hall of India Private Limited, 1994

James R. Munkres,Topology A First Course, Prentice-Hall of India Private Limited, 1994. 9

work page 1994

[6] [6]

Cohen,Set Theory and the Continuum Hypothesis, Dover Publications, Inc., 1994

Paul J. Cohen,Set Theory and the Continuum Hypothesis, Dover Publications, Inc., 1994

work page 1994

[7] [7]

Hocking and Gail S

John G. Hocking and Gail S. Young; Topology (Dover Publications, Inc., New York, 1961)

work page 1961

[8] [8]

Kobayashi and K

S. Kobayashi and K. Nomizu,Foundations of Differential Geometry:V ol.I, Wiley Classics Library, 1991

work page 1991

[9] [9]

Courant and F

R. Courant and F. John,Introduction to Calculus and Analysis:V ol.1, Springer-Verlag New York Inc., 1989

work page 1989

[10] [10]

Ghosh, International Journal of Pure and Applied Mathematics,76, No.2, pp.251 - 260, (2012)

K. Ghosh, International Journal of Pure and Applied Mathematics,76, No.2, pp.251 - 260, (2012)

work page 2012

[11] [11]

Itzykson and J

C. Itzykson and J. B. Zuber,Quantum Field Theory, Dover Publications, Inc. Mineola, 2005

work page 2005

[12] [12]

Tolman,The principals of Statistical Mechanics, Dover Publications, Inc., New York, 1979

Richard C. Tolman,The principals of Statistical Mechanics, Dover Publications, Inc., New York, 1979

work page 1979

[13] [13]

Apostol,Mathematical Analysis, Narosa Publishing House, 1992

Tom M. Apostol,Mathematical Analysis, Narosa Publishing House, 1992

work page 1992

[14] [14]

C. W. Misner, K. S. Thorne and J. A. Wheeler,Gravitation, W.H. Freeman and company, New York, 1970

work page 1970

[15] [15]

S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time, Cambridge University Press, 1973

work page 1973

[16] [16]

R. M. Wald,General Relativity, The University of Chicago Press, Chicago and London, 1984

work page 1984

[17] [17]

Lovelock and H

D. Lovelock and H. Rund,Tensors, Differential Forms, and Variational Principals, Dover Publications, Inc., New York, 1989

work page 1989

[18] [18]

Ghosh, Quantum Stud.: Math

K. Ghosh, Quantum Stud.: Math. Found.11, 625–642 (2024). https://doi.org/10.1007/s40509-024-00340-9

work page doi:10.1007/s40509-024-00340-9 2024

[19] [19]

Quigg,Gauge Theories of the Strong, Weak and Electromagnetic Interactions, Princeton University Press, Princeton and Oxford, 2013

C. Quigg,Gauge Theories of the Strong, Weak and Electromagnetic Interactions, Princeton University Press, Princeton and Oxford, 2013

work page 2013

[20] [20]

Aharonov and D

Y . Aharonov and D. Bohm,Significance of Electromagnetic Potentials in the Quantum Theory, The Physical Review. 115, No. 3, 1959, pp. 485-491

work page 1959