Exploring the Impossibility of Inaccessible Cardinality in Black Holes

The concept of inaccessible cardinality, a fascinating topic in set theory, brings us to an even more mind-bending question: What would happen if we attempted to put such a number of atoms into a black hole? While the answer is essentially 'impossible', let's delve into why this is the case and what the implications might be.

The Role of Event Horizons in Atom Disintegration

Atomic structures and the mass they possess are fundamental to our understanding of matter. However, when it comes to black holes, these concepts are fundamentally altered. The Apparent Event Horizon (AppEH) of a black hole is the boundary at which light and any other matter becomes unable to escape its gravity. Upon approaching the black hole, atomic structures are disintegrated, and the information we call mass begins to lose its meaning.

Much closer to the singularity, within the accretion disk, atoms and their constituents face an even more severe fate. Protons and neutrons are disintegrated into quarks, and even these particles are subject to further decomposition. What remains is a stream of elemental particles, such as electrons, photons, and quarks, falling into the black hole. These particles, albeit modified, are the closest most of them get to "falling into" the black hole.

Black Holes and the Essence of Infinities

Mentioning the word 'singularity' in the context of black holes immediately evokes images of extreme and often paradoxical conditions. Mathematically, a singularity is a point where the known physical laws break down. But from a physicists' perspective, the singularity itself is not a physical object. It is a mathematical construct that signifies the breakdown of the field equations. Any attempt to 'break' it further is considered a moot point, as it lies beyond the domain where our current understanding holds.

Given that a black hole's singularity is essentially an error in our mathematical models, it is not possible to put an infinite number of atoms into a black hole. If we were to somehow attempt to do so, the black hole would grow infinitely, but it would do so at a rate faster than the rate at which we could feed it, effectively preventing such an accumulation from occurring.

Theoretical Attempts at "Singularity Removal"

In light of the limitations posed by the singularity, another attempt to remove this mathematical anomaly involves theoretical models such as string theory. One such model, referred to as a "fuzzball", proposes that what we perceive as the singularity is actually a dense configuration of string-like objects. This model attempts to explain the geometry of the black hole's core without invoking a singularity.

Even with the string theory fuzzball model, attempting to inject a vastly larger number of atoms into the black hole would be theoretically impossible. The model posits a maximum density limit; even with an infinite number of atoms, their distribution and interaction with the string geometry would violate the model's rules. Injecting a larger quantity would be mathematically impossible, leading to a breakdown of string geometry and potentially resulting in a catastrophic event, perhaps even 'breaking' the universe as we know it.

Conclusion: A Resounding Impossibility

The implications of attempting to put an inaccessible number of atoms into a black hole highlight the limitations of our current understanding and the complexities of the universe. Whether via the breakdown of physical laws or the mathematical constructs of models like the "fuzzball", the attempt to achieve such a feat is fundamentally impossible. While the question remains intriguing, any real-world scenario cannot transcend the constraints of these theoretical models.