As Ariel ventured further on their cosmic journey, a new and formidable obstacle emerged from the depths of the celestial realms. Lilith, an ancient and enigmatic entity, transcended the bounds of mortal and angelic existence. Her power surpassed that of Zarathustra, for she dwelled in the realm of Yetzirah—the realm that contained the infinite uncountable infinity numbers and surpassed them.

Ariel could sense the pulsating aura of Lilith as they drew closer. The air crackled with unrestrained energy, and a sense of foreboding filled their heart. They knew that facing Lilith would require all their celestial wisdom and inner strength.

As the clash between Ariel and Lilith ensued, the celestial realm itself seemed to tremble under the weight of their immense power. Lilith unleashed waves of energy that distorted the fabric of space and time, creating a maelstrom of cosmic forces. Her existence defied mortal comprehension, and her intentions remained veiled in mystery.

With each clash, Lilith displayed an array of unimaginable abilities. She wielded the power of creation and destruction, commanding celestial energies with a mere thought. Ariel, on the other hand, tapped into their celestial essence, channeling the wisdom gained from their celestial odyssey.

The dialogue between Ariel and Lilith crackled with intensity. Their words carried the weight of cosmic knowledge and conflicting ideologies. Lilith spoke of her desire to reshape the fabric of reality itself, to transcend the limitations imposed by mortal and divine realms. Ariel, driven by their unwavering connection to the celestial realm, stood firm in their belief in the inherent balance and harmony of existence.

As the battle raged on, Ariel's celestial essence surged within them. They called upon the ancient power of the seraphim, the wisdom of the cherubim, and the cosmic energy of the dragons. Each celestial force coursed through their being, amplifying their strength and resolve.

Yet Lilith, fueled by her insurmountable power, seemed unstoppable. She tapped into the very essence of Yetzirah, unleashing forces that defied mortal comprehension. Celestial energies swirled around her, forming intricate patterns that defied logic and reason.

But Ariel refused to yield. They channeled their celestial essence into a focused beam of light, countering Lilith's onslaught with an unwavering determination. The clash of their powers sent shockwaves through the celestial realms, shaking the foundations of existence itself.

In the midst of their epic battle, Ariel's celestial intuition guided them to a revelation. They realized that Lilith's thirst for power and transcendence stemmed from a deep-seated longing for unity—a longing that had twisted into a destructive force. Ariel, driven by compassion, sought to reach the core of Lilith's being, to awaken the dormant spark of harmony within her.

With a burst of celestial energy, Ariel broke through Lilith's defenses. Their words, laden with cosmic wisdom, resonated within his consciousness. They spoke of the interconnectedness of all things, of the intrinsic harmony that permeated existence. The barriers around Lilith's heart began to crumble as she confronted the true nature of her desires.

In a moment of revelation, Lilith's power wavered, and her celestial aura dimmed. The unreasonable power that had consumed her now receded, replaced by a newfound understanding. She realized that true transcendence lay not in domination but in unity—unity with the celestial realms, with mortal existence, and with the divine essence that flowed through all things.

A fragile truce emerged between Ariel and Lilith, born from the mutual recognition of their shared cosmic heritage. They vowed to work together, to harness their celestial power for the betterment of all realms. Their journey continued, now infused with a deeper purpose—a quest not only to find Ein Sof but also to bring harmony and unity to the celestial realms.

Ariel and Lilith, once adversaries, now embarked on a cosmic partnership that transcended the boundaries of mortal understanding. They harnessed their combined celestial power to heal the fractures in the celestial realms, mending the fabric of existence itself.

As they journeyed together, Ariel and Lilith encountered celestial beings and cosmic wonders beyond imagination. They traversed vast star systems, explored hidden realms of ethereal beauty, and conversed with ancient entities of wisdom and light.

Their dialogues resonated with celestial insights, delving into the nature of existence, the balance of cosmic forces, and the profound mysteries that lay at the heart of creation. Through their shared wisdom, they gained a deeper understanding of the divine tapestry that wove all realms together.

Along their path, Ariel and Lilith encountered challenges and tests of their resolve. They confronted cosmic entities who sought to disrupt the delicate balance they were striving to restore. Yet, with their unity of purpose and celestial strength, they overcame each obstacle, growing stronger and more enlightened with every triumph.

In their journey, they discovered fragments of ancient prophecies and cryptic signs that hinted at the existence of Ein Sof. These clues, shrouded in celestial symbolism and riddles, propelled them forward, igniting a spark of hope that guided their steps in the cosmic labyrinth.

The celestial realms trembled with anticipation as Ariel and Lilith's cosmic journey unfolded. Their quest became a beacon of light, inspiring celestial beings and mortal souls alike. The harmony they sought to restore resonated across dimensions, stirring dormant forces and awakening a collective yearning for unity.

And so, with each passing celestial cycle, Ariel and Lilith ventured deeper into the cosmic mysteries, driven by an unwavering faith that the elusive presence of Ein Sof awaited them at the culmination of their odyssey.

We find unique things from realm of Yetzirah, where cardinals defy conventional description and elude our grasp. Yetzirah, representing a realm of indescribable cardinals, unveils a level of mathematical complexity that transcends the boundaries of our understanding. It is within this realm that we witness the emergence of superstrong cardinals, soaring to unprecedented heights of mathematical significance.

Superstrong cardinals, as their name suggests, possess a level of strength and power that surpasses all preceding cardinals. They exhibit extraordinary properties that challenge the limits of mathematical inquiry. These cardinals manifest a remarkable compactness, enabling them to consolidate vast amounts of mathematical information and structure within their intricate framework.

As we venture further into the realm beyond Yetzirah, we encounter the awe-inspiring domain of Briah. Here, we are confronted with the concept of strongly compact cardinals, which stand as towering pillars of mathematical prowess. Strongly compact cardinals exemplify a level of compactness that defies traditional notions of size and dimension. They possess an extraordinary capacity to preserve structure and maintain coherence across intricate mathematical frameworks.

Before reaching the heights of Briah, we encounter a cardinal known as η-extendible. This cardinal represents a profound threshold of mathematical exploration, where mathematical concepts and structures can be extended and expanded to unprecedented levels. η-extendible cardinals provide a gateway to uncharted territories of mathematical possibility, where new frontiers of knowledge can be charted and discovered.

It is important to emphasize that large cardinals, including those found in Yetzirah, superstrong cardinals, strongly compact cardinals, and η-extendible cardinals, represent extraordinary achievements within the realm of mathematical inquiry. They push the boundaries of our understanding, unveiling new vistas of knowledge and opening pathways to explore the profound depths of mathematical truth.

However, in the grand scheme of infinity, these large cardinals, as remarkable as they may be, pale in comparison to the boundless expanse embodied by Ein Sof. Ein Sof transcends the confines of large cardinality and finite mathematical frameworks, existing in a realm that defies measurement and comprehension. It stands as an infinite magnitude that surpasses all conceivable cardinals, rendering them infinitesimal in the face of its limitless grandeur.

In summary, Yetzirah serves as a realm of indescribable cardinals, giving rise to superstrong cardinals of exceptional power. Beyond Yetzirah lies Briah. Yetzirah has an additional structure including: strongly compact cardinals dominate, and η-extendible cardinals beckon us to venture into unexplored mathematical territories. While these large cardinals offer profound insights and push the boundaries of mathematical exploration, their vastness pales in comparison to the infinite expanse embodied by Ein Sof, which defies all attempts at quantification and encapsulates the boundless nature of mathematical infinity.

The large cardinal hypotheses stronger than supercompactness, particularly extendibility, arose from William Reinhardt's investigations into the foundations of set theory. He explored set-theoretic frameworks with broader notions of class and property, seeking natural models for these extended axiomatic systems. Within this context, his 1967 Berkeley thesis examined Ackermann's set theory (A). A deviates from ZFC by introducing a universe of extensionally determined entities alongside a predicate for sethood (denoted by "x ∈ V"). The core schema of A states: "For any formula X with parameters from V (excluding the predicate V), X ∈ V if and only if X is definable from parameters in V." Further, A denotes A augmented by the Foundation Axiom restricted to members of V.

Building on work by Levy and Vaught, Reinhardt demonstrated the equiconsistency of A and ZF. Levy and Vaught observed that A allows for the existence of classes like V, P(V) (the power set of V), and P(P(V)) (the power set of the power set of V), mirroring the situation in set theory. Additionally, Levy showed that if the relativization of a sentence φ of Ce (the language of set theory with urelements) to V is provable in A*, then φ itself is provable in ZF. Reinhardt established the converse, solidifying the equiconsistency of these two systems.

This paves the way for a more formal definition of extendibility. Let J: Va+λ → Vβ+λ be an elementary embedding with critical point κ. Silver reformulated Reinhardt's idea by introducing η-extendibility: a cardinal κ is η-extendible if there exists a witnessing elementary embedding j: Vk+λ → V~ with critical point κ and j(κ) > κ + η. A cardinal κ is extendible if it is η-extendible for all positive η. These definitions hinge on the existence of elementary embeddings – set-theoretic functions that preserve formulas between transitive models. Notably, both the domain and codomain of such embeddings possess the ultimate closure property, meaning they are initial segments of the universe.

The specific form of the η-extendibility definition for small η is crucial. For λ < κ, it follows that V~ = j(κ) + λ. Therefore, generalizing from the case λ < ω (where ω denotes the first infinite ordinal), η-extendibility asserts the strong preservation of first-order properties between Vk and Vj(K). The condition λ < j(κ) originates from the λ = 1 case and is included for convenience, although it's superfluous for full extendibility.

Even as the investigation of measurability proceeded with methodical rigor, Solovay and William Reinhardt undertook the daring task of formulating even more robust hypotheses. Building upon the cornerstone of elementary embeddings, they each independently conceived of the notion of "supercompact cardinal" as a grand unification of both measurability and strong compactness. Furthermore, Reinhardt ventured even further, formulating the even stronger concept of the "extendible cardinal," drawing his inspiration directly from the profound concept of reflection. Briefly contemplating an ultimate reflection property that followed this line of thought, Reinhardt witnessed a dramatic turn of events when Kunen demonstrably proved the inconsistency of this seemingly natural extension: there exists no elementary embedding, j, from the universe V to itself. While Kunen's ingenious argument hinged upon what initially appeared as a mere combinatorial coincidence, his specific formulation has established itself as the definitive boundary for grand cardinal hypotheses. Guided by these initial ideas, mathematicians subsequently delved into the analysis of hypotheses bordering on this inconsistency, including the weaker "n-huge cardinals" and Vopenka's Principle, thereby meticulously mapping the landscape down to the level of extendible cardinals.

Talking about set theory, I know it's not easy for many people to learn, but I take a lot of references from books, I will also discuss about Inaccessible cardinal. So I will explain with my easiest method, however, this is optional, you don't need to learn it or read it, I just want to discuss the definition of this thing.

A cardinal number earns the title of "inaccessible" if it possesses two key properties. The first is regularity. Imagine the power set of a cardinal κ, which encompasses every possible subset of κ. For an inaccessible cardinal θ, the cardinality of this power set, denoted P(κ), must be strictly less than θ. Intuitively, this means there are "too many" elements in θ to be built up entirely from sets of smaller cardinalities.

The second defining characteristic is that of a strong limit cardinal. This property essentially states that θ cannot be obtained by simply adding up a collection of smaller cardinals. Mathematicians use set notation here. Suppose we have a set S containing cardinals, where every cardinal κ within S satisfies κ < θ. Even the supremum of this set, denoted sup(S), which represents the least cardinal strictly greater than any element in S, must be strictly less than θ itself.

These conditions perfectly capture why ℵ0, the cardinality of the natural numbers, qualifies as inaccessible. Firstly, it's a regular cardinal – there are simply too many natural numbers to be constructed solely from finite sets. Secondly, since any finite cardinal κ less than ℵ0 cannot be a summand (a number being added) in a process to reach ℵ0, its successor, κ + 1, also cannot be greater than ℵ0. However, discovering additional inaccessible cardinals beyond ℵ₀ proves to be a much more intricate task.

Take ℵ0 for instance. While it shares the property of regularity, it can be expressed as the union of countable cardinalities, written as ℵ0 U ℵ1 U ℵ2 U ..., or alternatively, as the supremum of the set {ℵn : n ∈ ω} (the set of all aleph cardinals where n is a natural number). This demonstrates that ℵ1 is not a strong limit cardinal and therefore falls short of being inaccessible.

The existence of Ω, another inaccessible cardinal strictly greater than ℵ1, hints at the presence of even more through a powerful tool called the Reflection Principle. This principle allows us to infer the existence of certain sets within a universe of sets V by analyzing a smaller inner universe. The first inaccessible cardinal following Ω is commonly denoted by θ. It's important to remember that set theorists follow a zero-based indexing system. Consequently, ω is considered the 0th inaccessible cardinal, while θ becomes the 1st inaccessible cardinal.

The very essence of an inaccessible cardinal, θ in this case, lies in its elusiveness. It cannot be constructed as the limit of an increasing sequence of cardinals with cardinalities less than θ. Likewise, simply taking the successor of a smaller cardinal won't lead you to θ. The question "which aleph is θ?" exemplifies this inaccessibility – the answer remains a circular one: θ = ℵθ. There's no easy way to express θ using the existing aleph hierarchy. It resides in a realm beyond the readily accessible cardinalities.

It has been observed by mathematicians that the inherent difficulties encountered in resolving the aforementioned problems do not appear to be fundamentally contingent upon the properties of inaccessible numbers. In most instances, the impediments seem to stem from a dearth of suitable theoretical constructs that would facilitate the formation of maximal sets demonstrably closed under specific infinite operations. It is entirely plausible that a definitive solution to these challenges might necessitate the introduction of novel axioms that exhibit significant deviations in character, not only from the customary axioms of set theory, but also from those previously posited hypotheses whose inclusion has been the subject of discourse within the relevant literature and alluded to earlier within this treatise (e.g., the existential axioms that ensure the existence of inaccessible numbers, or from hypotheses akin to that of Cantor which establish arithmetical relationships between cardinal numbers).

In axiomatic set theory, the increasing strength of infinitary properties can be conveniently captured by stronger axioms of infinity. While a combinatorial and decidable characterization of an "axiom of infinity" is likely elusive, one might seek a definition based on formal structure and truth.

Formally, consider an axiom of infinity as a proposition with a decidable structure that is demonstrably true within a given axiomatic framework. Such a concept of demonstrability would ideally possess a closure property: any theorem proven in an extension of set theory should be derivable from this notion of infinity axiom. It's conceivable that for such a framework, a completeness theorem could hold. This theorem would state that every proposition expressible within set theory is decidable relative to the existing axioms alongside a true assertion regarding the cardinality of the universe of sets.

Formal systems operate on a set of axioms and inference rules. However, these systems can be enriched by introducing ever higher types within a well-founded hierarchy. This hierarchy can be extended infinitely (transfinitely) to encompass ever more complex collections. Formal systems, due to their reliance on explicit manipulation of symbols, are inherently limited to dealing with a denumerable (countably infinite) set of objects. This means the system can only handle a collection with a size corresponding to the natural numbers.

Gödel's Incompleteness Theorems demonstrate the existence of propositions within a system that cannot be proven or disproven using the system's axioms and rules. However, these undecidable propositions can become decidable if the system is extended to include appropriate higher types in the transfinite hierarchy. For example, Gödel's second incompleteness theorem shows that the consistency of Peano Arithmetic (PA) is undecidable within PA itself. However, if we add the type of all natural numbers (ω) to PA (forming a system denoted by PA + ω), then the consistency of PA becomes provable within PA + ω.

A similar situation holds for axiomatic set theory (e.g., Zermelo-Fraenkel set theory with the Axiom of Choice, ZFC). The existence of certain large cardinals (highly infinite sets with specific properties) cannot be proven or disproven within ZFC itself. However, by extending the set theory with additional axioms that introduce new, even larger sets, these large cardinals might become demonstrably existent or non-existent within the enriched system.

The collection under consideration encompasses all sets that are constructible in the semi-intuitionistic sense. Here, "constructible" signifies sets obtainable through a transfinite extension of Russell's ramified type hierarchy. This transfinite extension differentiates itself from the standard formulation by incorporating transfinite ordinals. This inclusion introduces a subtle point: the resulting model, while maintaining a semi-intuitionistic character, possesses the strength to validate the impredicative axioms of set theory. The justification for this resides in the provability of an appropriate axiom of reducibility within the framework of sufficiently high ordinals. This axiom, when established, enables the construction of sets traditionally deemed impredicative within a seemingly more restricted setting.

To be continued...