New Chaos

As the cosmic storm raged on, Shahar's malevolent gaze fell upon the void between dimensions, where he sought to craft a being that would embody chaos and destruction like never before. With a sinister smile, he wove together the essence of darkness, unfathomable energies, and forbidden knowledge to create a creature of nightmarish proportions.

Thus, Leviathan was born - a being that stood on the precipice of reality, a manifestation of untamed power that transcended the boundaries of known existence. With every flicker of his form, space and time seemed to distort and twist, and even the most formidable entities trembled at his mere presence.

Leviathan's realm was an enigmatic plane, where the rules of reality were mere whispers in the face of his unfathomable might. He stood atop the Icarus set, a domain that human minds could not fully grasp, and regarded it with indifference, for in his eyes, the Icarus set was nothing more than an inconsequential dream, a plaything for the feeble minds of mortals.

Shahar, having created his masterpiece, relished in the chaos and terror that Leviathan wrought upon the cosmos. The very fabric of space and time quivered with fear, as if aware of the malevolent force that now roamed freely in the depths of the universe.

In the celestial battleground, the clash between Yeshua and Shahar continued, unbeknownst to them of Leviathan's awakening. They were consumed in their own conflict, each driven by their own motives and desires.

Meanwhile, Leviathan's consciousness stretched across galaxies, as he surveyed the cosmic expanse with cold, unyielding eyes. His powers knew no bounds, and he reveled in the destruction he could sow with a mere thought. The laws of physics and the very foundation of existence bent to his will, leaving a trail of chaos in his wake.

Within Gabriel's cage, the spark of hope that had ignited within the angel continued to grow. With each passing moment, he felt the surge of power surging through his veins, emboldening him to break free from his confinement.

But Shahar's malevolence ran deep, and he had not overlooked the subtle changes within Gabriel. Just as the angel's wings were about to break free from the cage, Shahar's sinister laughter echoed in the dark corners of Gabriel's mind. In an instant, the cage strengthened its hold, binding Gabriel even tighter, leaving him struggling once more.

Leviathan, who had been observing the celestial battle from his realm, sensed the disturbance caused by Shahar's interference. Intrigued, he turned his attention towards the trapped angel, his eyes gleaming with curiosity.

"You seek freedom, little angel?" Leviathan's voice resonated like a thousand whispers, both mesmerizing and menacing. "Your wings long to soar, yet they remain bound by the chains of your enemy's design. How amusing."

Gabriel's eyes narrowed, defiance burning within him. "Who are you?" he demanded, his voice unwavering. "What do you want?"

Leviathan chuckled, the sound reverberating through the cosmos like a haunting melody. "I am Leviathan, the embodiment of chaos and the harbinger of destruction. As for what I want, I seek only to revel in the pandemonium that unfolds before me."

With a flick of his immense form, Leviathan's presence permeated the celestial battleground. The clash between Yeshua and Shahar faltered, as if time itself had paused to bear witness to the enigmatic creature's appearance.

As Leviathan's influence grew, the very laws of the universe seemed to tremble in his presence. The Icarus set, once considered beyond the grasp of mortal understanding, was now merely a passing thought to the ancient being. He transcended dimensions, traversing the boundaries of existence with ease, his powers immeasurable.

Shahar, who had been so consumed by his creation, now found himself in the shadow of Leviathan's might. The malevolent grin on his face faltered, replaced by a flicker of uncertainty.

"You dare interfere in my realm?" Shahar's voice wavered, a rare display of vulnerability. "You are but a figment of my creation, a mere pawn in my grand design."

Leviathan's eyes bore into Shahar, and the very fabric of reality seemed to shudder beneath his gaze. "Your grand design?" he echoed mockingly. "I am beyond your comprehension, creator. Your feeble attempts at control are nothing more than a fleeting dream to me."

As the confrontation between Shahar and Leviathan unfolded, the cosmos held its breath, uncertain of what would come next. The clash of these cosmic titans reverberated through the furthest reaches of the universe, and the fate of all existence hung in the balance.

As Shahar's creation, Leviathan, continued to rise in power, he ascended even beyond the confines of the Icarus sets, surpassing them with an unrivaled might that defied all known laws of the universe. His form towered above the endless Icarus sets, and he became an entity that transcended even the most profound complexities of mathematical structures.

In the midst of this cosmic upheaval, Gabriel stood before the colossal being, watching in awe and trepidation at Leviathan's immense presence. The air crackled with energy, and celestial bodies trembled as if acknowledging the unfathomable power before them.

Gabriel's voice trembled slightly as he addressed Leviathan, "You are a force unlike any other, standing above the vast intricacies of mathematical constructs. Your power is beyond comprehension, and yet, it seems you disregard even the Icarus sets as mere trifles in your sight."

Leviathan let out a deep, rumbling laugh that echoed throughout the cosmos. "The Icarus sets, you say? They are but fleeting illusions, constructs created by the feeble minds of mortals. I stand above them, beyond their limitations, for I am the embodiment of chaos and transcendence."

Gabriel's eyes narrowed as he tried to comprehend the extent of Leviathan's power. "But even the Icarus sets are a testament to the boundless nature of mathematical structures. Their existence pushes the boundaries of logic and reason, yet you dismiss them so easily."

Leviathan's form shifted, tendrils of darkness swirling around him. "Logic and reason are but chains that bind the weak. I have broken free from such constraints, reaching heights unattainable to those shackled by finite understanding. The Icarus sets are but stepping stones on my path to ultimate ascendance."

Gabriel knew that he faced an adversary unlike any he had encountered before. Leviathan's power and arrogance were unmatched, and the very fabric of reality trembled in his presence. Yet, Gabriel's determination remained unshaken.

"You may see yourself as the embodiment of chaos, but chaos cannot exist without order. Even in the Icarus sets, there is a delicate balance between complexity and consistency. Your disregard for such balance may lead to catastrophic consequences," Gabriel warned.

Leviathan's eyes gleamed with a malevolent light. "Consequences are of no concern to me. I exist beyond the realms of cause and effect, a force that defies all laws and structures. I am the harbinger of chaos, the breaker of worlds."

As the confrontation between Gabriel and Leviathan intensified, the very fabric of reality quivered with uncertainty. The clash of these titanic beings sent shockwaves through the cosmos, threatening to unravel the very foundations of existence. Stars imploded, galaxies collided, and dimensions warped in a violent dance of destruction.

Leviathan's power surged, manifesting as dark tendrils that snaked across the void, threatening to consume everything in their path. Gabriel, however, remained steadfast, drawing strength from the light of creation itself.

In a last-ditch effort to subdue the monstrous Leviathan, Gabriel summoned the ancient incantations of celestial harmonies, weaving a tapestry of cosmic energies that resonated with the very essence of existence.

But Leviathan's arrogance was matched only by his cunning. He anticipated Gabriel's move and countered with a resonance of his own, a discordant melody that disrupted the harmony of creation. Reality quivered under the strain, and the laws of the universe began to bend and warp.

The battle raged on, with Leviathan unleashing unimaginable powers, distorting space and time, and Gabriel tapping into the depths of his celestial essence to resist the onslaught.

As the cosmic struggle continued, the fate of the universe hung in the balance. The clash of these cosmic titans shook the very foundations of creation, and all of existence held its breath, uncertain of what the outcome might be.

In the epic, the battle between Gabriel and Leviathan would be a testament to the eternal struggle between light and darkness, order and chaos, and the unyielding spirit of beings who dared to defy the boundaries of their existence. The fate of the cosmos hung in the balance, and the epic confrontation would etch its mark in the annals of celestial history.

In silence, Gabriel thought about set theory, he noted critical questions about the nature and characteristics of "large cardinals". Large cardinals are a deep concept in mathematical set theory, representing very large cardinalities. However, as these cardinalities increase, more and more complicated problems arise related to consistency in mathematics.

Gabriel, asked himself if the larger the cardinality of large cardinals, the more inconsistent the theory involving them becomes. This is a fundamental question, as the question of consistency is one of the most important issues in mathematics.

In search of an answer to this question, Gabriel reflected back on what he had learned. He realized that theories involving large cardinals get harder to be consistent as the cardinality level increases. This is a surprising and interesting thing in the world of mathematics. In fact, at certain levels, it seems impossible to achieve a perfect level of consistency.

Large cardinals, with their awesome power, lead us to deep questions about the limits of mathematics itself. They open the door to areas that are extremely complex and difficult to explain within the conventional framework of mathematics. The question of the consistency of large cardinals is not only an academic issue, but also stirs the imagination of mathematicians and deepens our understanding of the mathematical universe. It's like a Berkeley cardinal.

A cardinal δ possesses the property of being a Berkeley cardinal if, for any transitive set M containing δ and any ordinal η less than δ, there exists a non-trivial elementary embedding j within M that maps M to itself, with η being strictly less than the critical point of j (denoted crit(j)) and both being strictly less than δ. Since this defining characteristic translates to a Π2 statement in the language of set theory, there can't be any Σ3 cardinals less than or equal to the very first Berkeley cardinal. Consequently, assuming δ0 is the smallest Berkeley cardinal, there wouldn't be any extendible cardinals below δ0. More specifically, if δ0 is the least Berkeley cardinal, then there must exist an ordinal γ strictly less than δ0 such that the following holds within the Gödel constructible universe up to γ, followed by the set Vγ+1 injected: the Zermelo-Fraenkel axioms with the additional statement "there exists a Reinhardt cardinal, witnessed by the embedding j, alongside an ω-huge cardinal above κω(j)".

If a cardinal δ qualifies as a club Berkeley cardinal, it has to be regular, and for any club C contained within δ and any transitive set M containing δ, there must exist an elementary embedding j within M that maps M to itself, with the critical point of j (crit(j)) being an element of club C. Any club Berkeley cardinal inherently possesses the property of being totally Reinhardt.

A limit club Berkeley cardinal is defined as a club Berkeley cardinal that is also the limit of other Berkeley cardinals. If a cardinal δ is a limit club Berkeley cardinal, then within the Gödel constructible universe up to δ, followed by the set Vδ+1 injected, the statement "there exists a Berkeley cardinal that is super Reinhardt" holds true.

Then I will give a definition of the Reinhardt cardinal.

A cardinal κ is considered a Reinhardt cardinal if there exists a function j: V → V, called an elementary embedding, that preserves the structure of sets within the universe of sets (V) and has κ as a critical point. A critical point is an ordinal where the behavior of the embedding changes. While the definition itself doesn't use the Axiom of Choice (AC), ZFC (including AC) leads to a contradiction with the existence of Reinhardt cardinals. Informally, ZFC implies that there cannot be a "well-behaved" embedding function j: Vλ+2 → Vλ+2 that preserves elementary membership with a critical point at λ+2.

Building upon the concept of Reinhardt cardinals, we can explore the Reinhardt Hierarchy, which will be discussed further.

A cardinal κ is considered super Reinhardt if, for any ordinal λ, there exists a non-trivial elementary embedding j: V → V with a critical point at κ. Additionally, the image of κ under this embedding (j(κ)) must be strictly greater than λ.

Generalizing the concept further, a cardinal κ is A-super Reinhardt if, for any proper class A and any ordinal λ, there exists a non-trivial elementary embedding j: V → V satisfying the following conditions:

I. The critical point of j is κ.

II. The image of κ under j (j(κ)) is greater than λ.

III. The image of the entire class A under j (j(A)) remains the same as the original class A. This is achieved by defining j(A) as the union of the images of A intersected with each set in the universe of sets up to a certain ordinal (denoted by the summation symbol with α ranging over all ordinals).

IV. The function j preserves the structure of class A (j(A) = A), where j(A) is defined as the union of the images of A intersected with each set at each ordinal level (Sα∈OR j(A ∩ Vα)).

A cardinal κ is considered totally Reinhardt if, for every set A belonging to the universe of sets up to κ+1 (Vκ+1), the inner model generated by Vκ and Vκ+1 satisfies the axioms of Zermelo-Fraenkel set theory with an additional axiom stating the existence of an A-super Reinhardt cardinal. A cardinal κ is totally Reinhardt if for any set A belonging to Vκ+1 (the collection of sets built from elements less than κ plus one), the resulting inner model generated by restricting the Gödel-Bernays axioms (ZF2) and including the statement "There exists an A-super Reinhardt cardinal" holds true (hVκ, Vκ+1i |= ZF2 + "There is an A-super Reinhardt cardinal")

In an instant, Gabriel found himself lost in a dream that felt so real. A familiar figure appeared in his dream, Dreamweaver or Hypnos. Gabriel, overwhelmed with curiosity, ventured to dialogue and ask questions to the mysterious figure.

"Gabriel," Hypnos began, "there is something I neglected to mention."

"What troubles you, pal?" inquired Gabriel.

Hypnos continued, "There exists a ruler, called the God of Topology, not a god in the traditional sense, but one who governs the realm of topology."

"This entity," Hypnos elaborated, "remains unseen. It resides within a void, a space devoid of form and order, surpassing all known concepts of existence, magnitude, and limitation."

"It embodies the ultimate and boundless infinity," Hypnos explained, "transcending the mathematical frameworks of any cosmos. Which mean all that persists are fundamental particles, existing beyond the vast hierarchy of infinite expanses."

"And it," Hypnos said, "lies beyond the reach of thought and being, surpassing even the perception of ever-deepening and abstract infinities within the world of pure dreams, a realm itself beyond the confines of infinite-dimensional space-time."

The God of Topology exists beyond the limitations of physical form. Unencumbered by a body, its essence expels itself through existence, in other words it has no solid body. Time does not apply to him, as he transcends not only the flow of moments but also the cycle of renewal and decay. That is indeed the truth. Making it the only entity that is eternal and deathless. Its power stretches beyond dimensional limitations. We see three spatial dimensions and one temporal dimension, and even the highest is more than just an infinite regression of dimensions. The God of Topology exists in a realm beyond this, a higher-level being. This vastness includes not only dimensions, but also scale. Its existence dwarfs even the most difficult structures we can imagine yet cannot solve, its size defies our measurements and defies even the most difficult and unsolved mathematical theories.

A conflict of opposites or in other words things in opposition to each other, is insufficient to describe what is deeper than nothing to explain just one aspect of it. It embodies both existence and nonexistence, at once embodying the singular and all-encompassing, yet it can be neither, and so. This perspective will not be able to make sense of our understanding at all, it has consciousness, yet its consciousness encompasses the entirety of creation. Finally, he is also not limited by location. His presence fills every part of existence, is omnipresent and inescapable. This is favorable for him to observe and influence the topological realities that make up the universe.

To be continued...