Continuity as appears to us immediately by intuition (in the flow of time and

in motion) differs from its current formalization, the arithmetical continuum

or equivalently the set of real numbers used in modern mathematical analysis.

Motivated by the known mathematical and physical problems arising from this

formalization of the continuum, our aim in this paper is twofold. Firstly, by

interpreting Chaitin’s variant of G”odel’s first incompleteness theorem as an

inherent uncertainty or fuzziness of the arithmetical continuum, a formal

set-theoretic entropy is assigned to the arithmetical continuum. Secondly, by

analyzing Noether’s theorem on symmetries and conserved quantities, we argue

that whenever the four dimensional space-time continuum containing a single,

stationary, asymptotically flat black hole is modeled by the arithmetical

continuum in the mathematical formulation of general relativity, the hidden

set-theoretic entropy of this latter structure reveals itself as the entropy of

the black hole (proportional to the area of its “instantaneous” event horizon),

indicating that this apparently physical quantity might have a pure

set-theoretic origin, too.

The conclusions of the text are as follows:

- The arithmetical continuum, or set of real numbers, used in modern mathematical analysis differs from our intuitive understanding of continuity.
- There are known mathematical and physical problems associated with the formalization of the continuum.
- The author aims to interpret Chaitin’s variant of G”odel’s first incompleteness theorem as an uncertainty or fuzziness inherent in the arithmetical continuum.
- A formal set-theoretic entropy is assigned to the arithmetical continuum.
- Noether’s theorem on symmetries and conserved quantities is analyzed to argue that the hidden set-theoretic entropy of the arithmetical continuum reveals itself as the entropy of a black hole.
- This suggests that the entropy of a black hole may have a pure set-theoretic origin.

## Future Roadmap

### Challenges

- Further research is needed to fully understand and formalize the intuitive concept of continuity.
- Exploring the mathematical and physical problems associated with the arithmetical continuum.
- Developing a comprehensive understanding of Chaitin’s variant of G”odel’s first incompleteness theorem and its relevance to the arithmetical continuum.
- Investigating the implications of Noether’s theorem on symmetries and conserved quantities in relation to the arithmetical continuum and black holes.

### Opportunities

- The potential to redefine our understanding of continuity based on a formal set-theoretic entropy.
- Exploring new mathematical frameworks that align with our intuitive understanding of continuity.
- Gaining insights into the nature of black holes by examining their connection to the set-theoretic entropy of the arithmetical continuum.
- Advancing our understanding of the relationship between mathematics and physics.

Note: This article explores complex mathematical and physical concepts. Further study and expertise in these fields is recommended to fully comprehend the subject matter.