Hello everyone, Son Raon here and I'm back with another cosmological chapter for Raonverse.

The reason why I wrote this chapter is mainly due to the fact that I know that some of you - that are either mathematically based scalers or fans of mathematics in general - may have some questions as to what types of Cardinals exist.

---------------------------------------------------------

In order to avoid any confusion or contradictory thoughts from both you and I in regards to what I've written up to know let me define what do I mean by Cardinals :

In the science of mathematics, cardinality or cardinal number is a number that defines a set of numbers.

These sets can be finite - where the cardinal number in this is a natural number (a.k.a 0,1,2,3... etc) - or flat out infinite where in the latter case infinite cardinal numbers have been introduced.

The smallest of these infinite cardinals are often denoted with the Hebrew letter Aleph (אָ) and a number that shows their order of magnitude in the infinite Aleph Cardinals.

These cardinals go all the way from Aleph Null (אָ0) which is the smallest replaceable uncountable infinity to Aleph Omega (אָω) which is denoted as the Hebrew letter Aleph and the lowercase version of the Greek letter omega (ω) which is still an uncountable infinity, but far from the largest possible infinity.

Now let me explain which part of the Raonverse Cosmology has Aleph Cardinals in it's scaling.

This part is the spatial components of the verse's "normal" dimensions or as I used to call them subdimensions for the sake of simplicity.

These sub dimensions are infinite in size and quantity, on top of having an Aleph Cardinal Gap/Hierarchy between them.

Meaning that an infinite set of them is basically an Aleph Omega hierarchy.

---------------------------------------------------------

Now let's go to the next cardinal number, Worldly Cardinals :

I'll start by the definition. A Worldly Cardinal is a cardinal - let's abbreviate it as k - such as the rank Vk is a model of ZFC Set Theory.

In the verse ZFC is the quote on quote "Axiom of Inaccessibles" I use in the verse so in other words the Worldly Cardinals exist as a further result of ZFC existing in the verse.

Now as I explained before the sub dimensions are essentially a proven Aleph Omega hierarchy of dimensions, but due to the simple fact that they're essentially... for lack of a better term pieces in the puzzle that's the normal dimensions.

This in other words makes one of the infinite set of Normal Dimensions into a Worldly Cardinal Hierarchy.

---------------------------------------------------------

Now let's go a step higher into the "food chain" of cardinals introducing the Inaccessible Cardinals.

Now that's kinda tricky thing to explain since these cardinals have four "sub categories" which are:

- Weakly Inaccessible

- Strongly Inaccessible

- a-Inaccessible

- Hyper-Inaccessible.

---------------------------------------------------------

Let's start with the simpler one which is the Weakly Inaccessible Cardinals.

For starters as Inaccessible Cardinals are any uncountable cardinals that don't come to existence from procedures such as the cardinal arithmetic (retraction, subtraction, multiplication, division etc) of smaller cardinals.

Specifically, one of this uncountable cardinals is a Weakly Inaccessible Cardinal when it's a limit cardinal that's neither a successor cardinal nor flat out a zero.

A successor cardinal is to the total set of these cardinals what's 1 to 0 in ordinal arithmetic, essentially the next step.

Weakly Inaccessible Cardinals exist in the Raonverse as Infinite Sets of Infinite normal dimensions.

---------------------------------------------------------

Strongly Inaccessible Cardinals on the other hand are limit Cardinals that can't be reached by normal powersets or in other words stacking sets upon sets of infinite smaller cardinals.

With Strongly Inaccessible Cardinals we get to the Mental plane since a singular Uni.... Correction in this case a planet in the mental plane holds R>F Transcendence over the countless sets of dimensions between it and the physical "embodiments" of planets or stars in the Raonverse.

---------------------------------------------------------

Now for the a-Inaccessible Cardinals.

Since the term a-Inaccessible Cardinals is ambiguous between mathematicians since there are a lot of contradicting theories and statements, I'll use the least contradicting one.

The least contradicting one is the following.

If we denote a as an ordinal and i as an Inaccessible Cardinal then if we denote a second ordinal b with b<a, the set of b-inaccessibles less than i is unbound in i.

In case it's confusing let me give you an example from the cosmology of the Raonverse.

As I said before a planet in the mental realm is beyond the transfinite sets of dimensions between it and the physical form of the planet, making it a 0-Inaccessible Cardinal since 0-Inaccessible Cardinals are Strongly Inaccessible Cardinals.

In this case an a-Inaccessible cardinal would be the infinity of planets in the smallest possible structure other than planets and stars in the mental plane.

In other words a solar system since solar systems in the Raonverse contain infinite planets and stars

---------------------------------------------------------

Now for the Hyper-Inaccessible Cardinals.

A Hyper-Inaccessible Cardinal is following this theory.

Denote i as Inaccessible Cardinal, then as Hyper-Inaccessible Cardinal we denote a cardinal that has the form of i-Inaccessible.

In this case i-Inaccessible is essentially a cardinal that's Inaccessible to a cardinal belonging into the set of Inaccessible Cardinals.

In layman's terms that's basically a Weakly Mahlo Cardinal.

A Mahlo Cardinal is a Cardinal number that needs to fulfill the following requirements :

- Let's say that o is a limit ordinal number

- The Set of ordinals smaller than o is less than m is stationary in m(in other words, a non zero subset of the m cardinal)

- The m cardinal is a Mahlo Cardinal according to Paul Mahlo, the founder of the Mahlo Cardinal

The smallest possible solar system in the Mental Realm scales to it since it's basically a literal fractal loop essentially a mathematical structure that no matter how much you zoom in or out it extends ad Infinitum.

---------------------------------------------------------

Now the strongly Mahlo Cardinals. Similarly to the Strongly Inaccessible Cardinals, Strongly Mahlo cardinals are Mahlo Cardinals that can't be reached by continuous stacking of powersets.

In the Raonverse, especially in the mental plane (but it extends to any other parts of the verse in general) the solar systems that exist there have R>F Transcendence between them, giving them a strongly Mahlo Cardinal Hierarchy.

---------------------------------------------------------
a-Mahlo Cardinals are abiding under the laws of the same theory as a-Inaccessible Cardinals so I won't elaborate further on.

a-Mahlo Cardinals hierarchy exists in the mental plane since Galaxies formed by infinite Solar Systems exist in the Raonverse.

---------------------------------------------------------

Hyper-Mahlo Cardinals follow the same exact theory as the Hyper-Inaccessible Cardinals so essentially a Hyper-Mahlo Cardinal is essentially a Weakly Compact Cardinal.

Formally a Weakly Compact Cardinal is a Cardinal number made from the mathematicians Erdős and Tarski in 1961, trying to define a Weakly Non Compact Cardinal , but due to some implications in their theory, the name changed into Weakly Compact Cardinal.

The theory is as follows :

For a cardinal to be Weakly Compact, they have to be uncountable and for every mathematical function f:[k]² -> {0,1} there's a cardinal set of k homogeneous - in other words the same as - to f.

In this context we have:

- [k]² means the subset of 2-element subsets of k

- a subset of k is homogenous to f if and only if all of [S]² maps to 0 or all of [S]² maps to 1

Does that [S]² seem familiar to you?

In case you don't remember it , it's basically a reference to the Hyper-Inaccessible Cardinals and Hyper-Mahlo theory.

A Weakly Compact Cardinal Hierarchy in the Raonverse is the fact that those fractal looped galaxies have R>F Transcendence between them and since those R>F Transcendences happen in Fractal Loops that extend ad Infinitum, we can comfortably say that their cardinality extends to the highest extensions of Mahlo Cardinals.

---------------------------------------------------------

Now let's get to the juicy stuff, Woodin Cardinals.

Woodin Cardinals were cardinal numbers created by none other than Hugh Woodin and follow the following theory.

λ is Woodin if and only if λ is strongly inaccessible and for all
Α ⊆ Vλ there exists a λA < λ which is <λ-A-strong.

λA being <λ-A-strong means that for all ordinals α<λ, there exist a j:V\to M which is an elementary embedding with critical point, λΑ, jλΑ > a, Va ⊆ M and j(A)∩Va = A∩Va. (See also strong cardinal.)

Woodin Cardinals exist in the verse up to their highest point in the R>F Transcendant galaxies of the Raonverse Cosmology.

Although the Woodin Cardinals are incompatible with the Axiom of Choice due to the fact that the Axiom of Determinacy that Hugh Woodin used as a "stepping stone" for their creation isn't completely compatible with a well ordering of the real numbers as shown by the Axiom of Choice.

This isn't an issue in the verse due to the usage of External Modal Realism, the Theory that I've been mentioning in previous chapters that turns any contradictions to irrelevant since in Extended Modal Realism all impossible = all possible due to the fact that normally, impossibilities are essentially possibilities with either negative or zero percentage of becoming true, something that's rendered irrelevant due to the usage of the EMR.

---------------------------------------------------------

Now let's get to the next one. The Axiom of Wholeness.

Now you may ask, what the heck does the Axiom of Wholeness do in the Cardinals when it's an Axiom?

It's quite simple, it's an Axiom used for the high end cardinals and cardinal properties. More specifically the wholeness axiom states roughly that there is an elementary embedding j from the Von Neumann universe V to itself.

Where an elementary embedding of a structure N into a structure M of the same signature σ is a map h: N → M such that for every first-order σ-formula φ(x1, ..., xn) and all elements a1, ..., an of N,

N ⊨ φ(a1, ..., an) if and only if M ⊨ φ(h(a1), ..., h(an)).

And V is a Von Neumann Universe, a.k.a a Transfinite set of cardinals.

This has to be stated carefully to avoid Kunen's inconsistency theorem stating (roughly) that no such embedding exists.

---------------------------------------------------------

Before I mention High End Cardinals, I'll have to mention another set of Axioms known as the Rank Into Rank Axioms.

The Rank Into Rank Axioms has four parts that are symbolised from the smallest possible to the highest possible as I3, I2, I1, and I0.

I'll give you a brief description of each one in the following separate paragraph along with the corresponding structure of the Raonverse that's connected to everyone of them.

---------------------------------------------------------

The Rank Into Rank Axiom 3 also abbreviated as Axiom I3 is a branch of Rank Into Rank Axiom that states the following:

There's a non trivial elementary embedding of Vλ towards itself.

The Rank Into Rank Axiom 2 also abbreviated as Axiom I2 is a branch of Rank Into Rank Axiom that states the following:

There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ where λ is the first fixed point above the critical point.

The Rank Into Rank Axiom 1 also abbreviated as Axiom I1 is a branch of Rank Into Rank Axiom that states the following:

There is a nontrivial elementary embedding of Vλ+1 into itself.

The Rank Into Rank Axiom 0 also abbreviated as Axiom I0 is a branch of Rank Into Rank Axiom that states the following:

There is a nontrivial elementary embedding of L(Vλ+1) into itself with critical point below λ.

The structures of the Raonverse that follows the Rank Into Rank Axiom are the fractal looped, R>F Transcendant Galactic Clusters

---------------------------------------------------------

Beyond this we find the first quote on quote "problem" with our journey within the Cardinals is the Kunen's Inconsistency Theorem.

The Kunen's Inconsistency Theorem is the following:

In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen (1971), shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.

Some consequences of Kunen's theorem (or its proof) are:

- There is no non-trivial elementary embedding of the universe V into itself. In other words, there is no Reinhardt cardinal.

- If j is an elementary embedding of the universe V into an inner model M, and λ is the smallest fixed point of j above the critical point κ of j, then M does not contain the set j "λ (the image of j restricted to λ).

- There is no ω-huge cardinal.

- There is no non-trivial elementary embedding of Vλ+2 into itself.

This Theorem outright refutes the existence of higher cardinals which would be a problem within the verse itself since everything we've talked about up to now would essentially be debunked to oblivion.

But fear not.

Two "friends" will come to the rescue.

Those friends of ours are Extended Modal Realism and the V=Ultimate L Theorem.

Extended Modal Realism basically debunks Kunen's Inconsistency Theorem since it allows no contradictions or Impossibilities to exist.

The second friend of ours is the V=Ultimate L Theory that although impossible to be proven in reality, it's not impossible due to the fact that there's the existence of EMR that makes its proof into a possibility.

If you want to see more about the V=Ultimate L Theorem, please refer to this video from a seminar on Cardinality from Hugh Woodin himself on the topic of the V=Ultimate L Conjecture :

https://www.youtube.com/watch?v=YZnV8Y6Vc7Q

---------------------------------------------------------

Then we have the "big guns" of Set Theory which are:

- Reinhardt Cardinals
- Berkeley Cardinals
- 0=1 Axiom
- Icarus Axioms
- Explosion Principle
- Cantor's Theorem
- Cantor's Attic
- Cantor's Absolute Infinity.

---------------------------------------------------------

Reinhardt Cardinals are Cardinal Numbers that are following this theory :

A Reinhardt cardinal is the critical point of a non-trivial elementary embedding j : V→ V into itself.

This definition refers explicitly to the proper class j. In standard ZF, classes are of the form
{x|φ(x,a)} for some set a and formula φ. But it was shown in Suzuki (1999) that no such class is an elementary embedding j : V→V. So Reinhardt cardinals are inconsistent with this notion of class.

There are other formulations of Reinhardt cardinals which are not known to be inconsistent. One is to add a new function symbol j to the language of ZF, together with axioms stating that j is an elementary embedding of V, and Separation and Collection axioms for all formulas involving , but I'm not going to explain them since I wouldn't want this chapter to reach over a billion words due to the metric ton of theorems, Axioms and conjectures this chapter will have within it.

The Reinhardt Cardinals, up to their highest point - a.k.a the Totally Reinhardt Cardinals - are forming the fractal looped, R>F Transcendant, Galactic Superclusters.

I know that their inconsistency with ZFC may be a problem for their existence, but it's essentially the multiple times proven EMR what's hard carrying the verse up to now.

Not all heroes wear capes, some of them wear words and equations. Some of them are brilliant scientists of days long gone wearing the masks of the theorems they created.

---------------------------------------------------------

Now let's go to the highest known Cardinals in Set Theory. The Berkeley Cardinals.

The Berkeley Cardinals are Cardinal numbers that follow the according Theory :

In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992.

A Berkeley cardinal is a cardinal κ in a model of Zermelo-Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial elementary embedding of M into M with α < critical point < κ.[1] Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice.

A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that we have elementary

j1, j2, j3, ...
j1: (Vκ, ∈) → (Vκ, ∈),
j2: (Vκ, ∈, j1) → (Vκ, ∈, j1),
j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2),
and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice.

While all these notions are incompatible with Zermelo-Fraenkel set theory (ZFC), their
Π{2}^{V} consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example:

For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.

Now the Berkeley Cardinals also have three subcategories which are:

- Berkeley Cardinals
- Club Berkeley Cardinals
- Limit Club Berkeley Cardinals

This is good since there are three adjacent parts of the Mental Realm that can prove these cardinals.

---------------------------------------------------------

The Berkeley Cardinals are the elephant in the room in this occasion so I'll explain to you their connection points to the cosmology quite simply.

I've told you before that the Raonverse Cosmology is a composite cosmology for two reasons:

- Firstly because I've connected all my stories to it so that the characters would have an as close to evenly matched encounters as possible in future crossovers for both narrative and scaling reasons since it'd be difficult to keep a track of the cosmology of each story if I made a specific one for each story.

- Secondly, because I've stated before that the cosmology itself is an amalgamation of many cosmological structures, models and constants with a variety of implications for the verse.

Now in case I managed to confuse you more than before allow me to stop confusing you by explaining the Schwartzchild Cosmology also known as Black Hole Cosmology.

Black Hole Cosmology was a cosmological theory proposed by the theoretical physicist Raj Pathria and concurrently by the mathematician I.J.Good in 1964.

This cosmological theory has as basis the Theorem that if the Hubble Radius of the observable universe is the same as the Schwartzchild Radius according to the Schwartzchild Proportionality Constant then it'd come as a result that the observable universe is nothing more than a simulated Universe within the literal and metaphorical bowels of a black hole.

This theory is impossible to be proven due to the lack of sophisticated technology that we have in real life so many cosmologists think about it as nothing more than a coincidence.

Fortunately enough for the Raonverse this "coincidence" is what's necessary in order to be proven since first of all, the Raonverse is a fictional verse so essentially I as the author can have some.... for lack of a better term creative liberties about it.

Secondly it's because no matter if it's the real world or a fictional world there's a sertain saying that all have to remember.

A saying by Markus Aurelius, the stoic Roman Emperor about the coincidences that goes like this:

"There's no luck or coincidence in this world.

What people call luck or coincidence is the incredible result that happens when preparation meets opportunity"

In our case the preparation is the multiple times proven Extended Modal Realism that turns Impossibilities to possibilities whilst the opportunity is the Black Hole Cosmology.

If your way of thinking is similar to mine then the results would be simple.

The Berkeley Cardinals are essentially the "Observable Universe" within one of the Transfinite black holes in the Raonverse.

The Club Berkeley Cardinals are their respective fractal looped galaxies.

The Limit Club Berkeley Cardinals are the R>F Transcendence between said galaxies.

---------------------------------------------------------

Then I'll explain the Icarus Sets, the Explosion Principle and the 0=1 Axiom since they don't have that much to their name.

Let's start with the Explosion Principle. The Explosion Principle or Principle of Explosion is defined as follows:

In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]'), or the principle of Pseudo-Scotus (falsely attributed to Duns Scotus), is the law according to which any statement can be proven from a contradiction.[1][2][3] That is, from a contradiction, any proposition (including its negation) can be inferred from it; this is known as deductive explosion.[4][5]

The proof of this principle was first given by 12th-century French philosopher William of Soissons.[6] Due to the principle of explosion, the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous; since any statement can be proven, it trivializes the concepts of truth and falsity.[7] Around the turn of the 20th century, the discovery of contradictions such as Russell's paradox at the foundations of mathematics thus threatened the entire structure of mathematics. Mathematicians such as Gottlob Frege, Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem put much effort into revising set theory to eliminate these contradictions, resulting in the modern Zermelo-Fraenkel set theory.

If you didn't understand it, I'll give you an example. Let's use as example two statements "stars aren't only yellow" and "stars are only yellow" and let's suppose that both are true so therefore any other statements such as "dragons are real" are true with the following argumentative logic:

1. Stars aren't only yellow is true

2. Stars are only yellow is also true since we've stated that both are true.

3. Therefore the two-phase statement Stars are only yellow or dragons are real is also true by this logic since the former part is both true and false.

4. Since we know that the first part - Stars are only yellow - isn't true as it was falsely assumed of being true, then according to the Principle of the Explosion, the second part would have to be true so that the two-phase truth is true.

In a different solution to these problems, a few mathematicians have devised alternative theories of logic called paraconsistent logics, which eliminate the principle of explosion. These allow some contradictory statements to be proven without affecting other proofs.

This is the only thing needed in order to remove the nuisance of the Principle of Explosion out of our minds due to the usage of EMR.

---------------------------------------------------------

Now I shall explain both of the 0=1 Axiom and the Icarus Set since they're basically the same thing.

The Icarus Set was named that way from the ancient Greek myth of Daedalus and Icarus uprising against the King of Crete, Minos.

King Minos specifically had kept Daedalus and Icarus on the island so that they could construct for him a nigh inescapable labyrinth where he would be able to house the Minotaur, the living weapon/symbol of the tyranny of Minos, the latter saying falsely that it was just a way to impirison Minotaur in the labyrinth so that the Minotaur wouldn't hurt anyone.

Unfortunately for Daedalus and Icarus though, after they made the plans for Mino's labyrinth, the king had imprisoned them in a cell on the top of a mountain. The only exit was a dangerous cliff towards the sea.

This though didn't stop Daedalus and Icarus from escaping since they managed to escape by creating wings made out of seagull feathers, wax, leather, wood and rope.

Once the wings were ready, they were capable of flying above the cliff. Both father and son would have returned back to their home safely if it wasn't for the son flying to close to the sun, to the point of the wax on his wings melting off rapidly, having the son to plummet to his death in the sea beneath him, the sea being known, is known and will be known as the Icarian Sea.

In case you didn't understand the allegory of the myth, let me explain it to you.

Daedalus is the personification of the current mathematicians.

The Labyrinth is the Set Theory.

The Minotaur is the totality of the large Cardinals.

The Cell is the inconsistencies on their path.

The cliff is the failures the scientists were going to have.

Icarus is the Icarus Set else known as 0=1 Axiom.

The sun is Cantor's Absolute Infinity.

Essentially the Icarus Set is also known as the Sui**de Set since it comes too close to the metaphorical sun that's the Absolute Infinity, but fails to grasp and comprehend it.

That's one of the final frontiers before the end of Set Theory and above.

Another frontier that's surpassed effortlessly with EMR.

---------------------------------------------------------

Now in the end of the chapter we have the triple C Theorems which are:

- Cantor's Paradox
- Cantor's Attic Theorem
- Cantor's Absolute Infinity.

Cantor's Paradox essentially says that there isn't any upper limits of the Cardinals which in this verse this is a blessing and a curse.

A blessing because a universe in the Raonverse is practically beyond transfinite in terms of mathematics.

A curse in a sense that the verse extends more than the amount of currently known Cardinals which when you think about it, this paradox for any verse that has theories like EMR at place is more of a blessing than a paradox.

---------------------------------------------------------

Now we get to the juicy stuff, that are Cantor's Attic Theorem and Cantor's Absolute Infinity.

Cantor's Attic Theorem is proven as follows :

Imagine an Attic In A mathematical home.

An Attic that's composed of three parts :

The Lower Attic where all ordinals exist.

The Middle Attic where all "normal" Cardinals exist

The Upper Attic where all High End Cardinals exist.

This is pretty much what we've explained up to now.

---------------------------------------------------------

Now comes the "final boss" of this Set Theoretical rodeo, the Absolute Infinity.

The Absolute Infinity of Georg Cantor is proven as follows:

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite.

The only place in the Mental Realm that's proper for the magnum opus of mathematical theories are the Algebraical Dimensions. The concept of mathematic personified.

---------------------------------------------------------

Now comes the fun part since in the Mental Realm all philosophies exist as well and due to the fact that this Realm is an abstract Realm. This means that the philosophical absolute infinity exits as well.

Also let's not talk about the fact that there are transfinite Algebraical Dimensions since the scaling wouldn't end that easily.

I hope you'll all enjoy reading this chapter as much as I enjoyed writing it.

Number of Words : 4439 Words