Attributes Skyrocket Infinitely, I Dominate the Multiverse Chapter 700 Traveling Through All the "Worlds", Extreme Berkeley (1/3)

In fact, although they are all meaningless sources.

But the source in the "Second World" where Mu Cang is now located is far beyond the "First World" [End of Number] level in terms of overall strength... or it can be called a higher-order source of the Reinhardt cardinality level.

And the large cardinality corresponding to the unknown equal-order anomaly strength of this meaningless source is... the special-complete Reinhardt cardinality.

If you want to understand this large cardinality, you have to start with the super Reinhardt cardinality.

The so-called super Reinhardt cardinality, as the name suggests, is a super-high-order enhanced version of the Reinhardt cardinality.

So in essence, it is also a critical point of non-trivial basic embedding, embedded in itself.

At the same time, between these two large cardinality, there is actually a Reinhardt cardinality defined by the n-order set theory formula set.

However, since the consistency strength of this large cardinality is far inferior to that of the super Reinhardt cardinality, it will be omitted for the time being.

In short, the specific definition of the super Reinhardt cardinality is:

There is an ordinal k, for each ordinal a, if there is a basic embedding j:V→V such that j(k)＞a, and k is a critical point of j, then k can be called a super Reinhardt cardinality.

Similarly, if k is a super Reinhardt cardinality, then there will be γ＜k, so that (5γ, Vγ+1) is a model of ZF?+Reinhardt cardinality existence axiom.

The ZF? in it can be understood as a second-order ZF axiom system.

Yes, the ZF system has first-order, second-order, third-order, fourth-order, and even more orders.

In general, compared with the Reinhardt cardinality, the super Reinhardt cardinality adds a limiting condition on its basis:

That is, j(k) must be large enough to meet expectations.

If the so-called "expectation" concept is elaborated, it means that for all ordinals a, j(k)＞a must be.

To further elaborate, the definition of the super Reinhardt cardinality involves transcendence over all ordinals.

That is, for any given ordinal a, a basic embedding can be found so that k is mapped to a larger ordinal.

In comparison, the Reinhardt cardinality only requires the existence of a basic embedding j:V→V such that k is a critical point of j, but does not require that j(k)＞a for all ordinals a, but the super Reinhardt cardinality is completely the opposite.

So the consistency strength of the latter is far... far better than the former.

But such a huge super Reinhardt cardinality is still far... far weaker than the Berkeley cardinality.

There is absolutely no comparability.

Therefore, it is necessary to look for a large cardinality with greater consistency strength in the higher-level "mathematical world".

That is, A-super Reinhardt cardinality.

Its specific definition is: for a suitable class A, if all ordinals λ have a non-trivial elementary embedding j: V→V, crt(j)=k, j(k)＞λ, and j?(A)=j(A)(j?(A):=U(a∈ord)j(AnVa), then such k can be called A-super Reinhardt cardinality.

In general, this large cardinality is equivalent to an advanced and enhanced version of the Reinhardt cardinality - an advanced and enhanced version of the super Reinhardt cardinality.

It is a greater generalization or It is said to be extended, so the gap between the two is so huge that it is simply indescribable.

But even so, even if it is so huge, the A-super Reinhardt cardinality is still far...far weaker than the Berkeley cardinality.

So we must use it as a stepping stone, leap up endlessly, and go to a higher level to find a higher order and larger cardinality.

That is, the complete Reinhardt cardinality.

Regarding the definition of this large cardinality, if we simplify it, it is:

If for every A∈Vk+1, there is (Vk, Vk+1)

Is ZF?+A-super Reinhardt cardinality exist axiom model, then such k is the complete Reinhardt cardinality.

So, can the strength of the complete Reinhardt cardinality surpass the Berkeley cardinality?

Unfortunately, it still cannot.

Because these two large cardinality cannot be clearly compared.

Or to put it more precisely, the difference in consistency strength between the two cannot be determined.

There is no way to know which of the two large cardinality will be stronger, and we can only roughly assume that the two can be marked with a slightly vague "=" sign in terms of strength.

So, can it really surpass the Berkeley cardinality in terms of consistency strength? What is the largest cardinality?

The answer is the special-complete Reinhardt cardinality.

Or it can be called... unbounded closed Berkeley cardinality.

Yes, the two completely different Reinhardt cardinality spectrum and Berkeley cardinality spectrum, after rising to a very high level, will actually have some mysterious fusion, and then become one.

This may be the magic and beauty of mathematics.

As for the so-called unbounded closed Berkeley cardinality that completely surpasses and overrides the complete Reinhardt cardinality and Berkeley cardinality in strength, its specific definition is in short:

If k is regular and for all unbounded closed sets c?k and all transitive sets of k∈, there are j∈e and crt(j)∈c, then such k can be called an unbounded closed Berkeley base.

And when you reach this level, there is a question worth asking.

That is, in the unbounded closed Berkeley basis-->>

On top of the numbers, are there even greater members of the Berkeley cardinality spectrum?

The answer is, there will be.

Generally speaking, if k is both a limiting Berkeley base and an unbounded closed Berkeley base, then k can be called a limit unbounded closed Berkeley base.

If we expand on this, it will be an extremely long and complex mathematical theory.

So if we want to give a brief explanation again, we have to start from the beginning of the Berkeley cardinal genealogy:

As we all know, the smallest Berkeley base cannot be a super Reinhart base, so on this basis, there is an interesting question:

That is, are there some Berkeley bases that can become super Reinhardt bases?

In order to answer this question, we can introduce the concept of a Berkeley base that is powerful enough to be both a rank-reflective Berkeley base and a super-Reinhardt base.

Specifically, if a base δ is both an unbounded closed Berkeley base and a limit of a series of Berkeley bases, then δ can be considered to be a limit unbounded closed Berkeley base.

The related theorem is:

If δ is the limit unbounded closed Berkeley base, then (Vδ, Vδ+1) is the model of the axiom "There is a Berkeley base that is a super Reinhardt base".

From this it can be concluded that:

For all transitive sets that satisfy 5δ+1∈ and all d∈δ and d is an unbounded closed subset of δ, there will be k∈d, and then for all a<δ, there will be j∈e, and finally such that (1)crt(j)=k; (2)j(k)>a.

But today, Mu Cang doesn't have much interest or thought in the so-called extreme unbounded closed Berkeley cardinality, which is too far away and is a large cardinality level.

Because, he is still just a Reinhardt base level life form.

Therefore, the only thing Mu Cang wants to do now, and what is necessary, is to replicate his behavior in the "first world".

In other words, this meaningless stream in front of Duo She can be called version 2.0.

The so-called mind moves and the body moves.

Mu Cang immediately activated the four heaven-defying magic skills and began to solemnly seize the body of the meaningless source 2.0 whose strength was equal to the special-complete Reinhardt base...or the unclosed Berkeley base. "Work"

.

Buzz——

Perhaps it is because the four divine skills have now shown initial signs of integration, so each power has obviously made breakthrough progress.

Therefore, in just an instant, this meaningless source 2.0 was completely taken over by Mu Cang and became one of his clones.

At the same time, Mu Cang's own life and strength level, under the combined effect of [Infinite Secret Strategy] and [Domination Invasion], jumped from the Reinhardt base level to the infinite level without any hindrance. The level is firmly stationed at the unbounded closed Berkeley cardinal level.

At this point, Mu Cang surpassed all previous achievements and officially became an unbounded closed Berkeley cardinal-level life form.

boom--

Afterwards, Mu Cang, who possesses such terrifying levels of power, just turned his thoughts and summoned all the Von Neumann Universe V, Gödel and other powerful and weak ones in the "second world". Constructable universe L, well-ordered universe o, well-founded universe F, special complex universe, set theory multiverse, complex complex universe, V-logic logical multiverse, Grothendieck universe?, ultimate L internal model universe...etc. When all the realms of numerology, as well as all those who are like masters, those who are undecided, the indigenous people of the big cardinal number, and the super-finite and infinite beings are all sucked away in the air, leaving only an empty, lonely and desolate super-type or It can be called the super super type "worldly".

But after doing all this, Mu Cang did not get the results he wanted.

"Sure enough, there aren't even any fragments left."

Shaking his head in disappointment, Mu Cang crossed the entire "world" in one step and returned to the omniscient tower, and began to move towards the next level of "the world".

…

In that omniscient tower that has no top, no bottom, no distance, no nearness, Mu Cang has been traveling for a long time.

During this period, the total number of "worlds" He has passed through has exceeded one million trillions.

But in these trillions and trillions of "worlds", Mu Cang has never discovered the third source of meaninglessness again.

Moreover, the overall intensity of these hundreds of trillions of "worlds" is actually far weaker than the "second world", and even far weaker than the "first world".

Among the "worlds" with the highest intensity, the strength of the strongest ones at the top has only reached the super-compact cardinal level, which is roughly equivalent to the ultimate L inner model universe.

This made Mu Cang suspect that the so-called "first world" that he had taken for granted before... was probably not the real first "world".

He even suspected that this omniscient tower might not have the so-called first "world" at all, nor the so-called tower foundation.

In other words, no matter which layer of the "world" there is, it is likely that it is just a certain layer of the tower of omniscience.

And there is no iron-clad rule that the next level will inevitably be more powerful than the previous level.

Therefore, Mu Cang guessed that this high tower might not have a base, a spire, or other similar structures at all.

In other words, any level of "the world" may be its base and spire, or a certain layer of the tower body.

Therefore, if Mu Cang and others move forward in a super logical structure with no beginning or end, they will never reach the end.

However, Mu Cang, who has the patience that can be called the unbounded closed Berkeley cardinal level, will not feel any irritation at all because of this.

Mu Cang has already felt it. He feels that as long as he can climb to a certain large base level that is not "too far" from him, he can completely integrate the four divine skills and give birth to new heaven-defying skills.

As for this level, I tend to think that... it should be the ultimate unbounded closed Berkeley base. <-->>

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As long as he reaches that level, the dilemma he faces will be easily solved.

Mu Cang is confident about this.

…

A long, long time passed, until Mu Cang had already passed through the infinite "world".

But among so many "worlds", there is not even one existence that can surpass Him in strength.

The strongest among them are just some other seats.

Just a stream of nonsense at the Berkeley radix level.

So Mu Cang could only continue walking towards the next unknown "world".

…

A long, long, long time passed again, so long that Mu Cang had already passed by... the unbounded closed Berkeley cardinal "world".

In so many "worlds", Mu Cang still did not find a second existence at the base level of the Unbounded Closed Berkeley with him, nor did he find any final fragments.

So Mu Cang could only continue walking.

But perhaps after finally walking enough, He, who was in the tower of omniscience, actually clearly sensed the world that was far away...far beyond the unbounded closed Berkeley cardinal level in the first layer of "world" he passed through at a certain moment. Horrible fluctuations.

Moreover, this magnificent fluctuation that made Mu Cang suddenly feel insignificant actually carried strands of extremely twisted and unusual logical power.

So the answer is obvious.

This is definitely another higher-order source of meaninglessness.

That is, the ultimate unbounded closed Berkeley base level source flow.

"After traveling through thousands of mountains and rivers and seeing all the prosperity of "the world", I finally..."

Mu Cang first sighed, then laughed and said, "I finally... found you!"

Buzz——

While laughing, Mu Cang suddenly stepped out of the tower of omniscience and re-entered the "world", and stepped across the boundless and immeasurable abyss of loss, passing through the infinite and infinite realm of logic at all levels.

Finally, he stopped at a higher-level source that was far larger and majestic than all the meaningless sources he had experienced along the way.

Immediately, Mu Cang's seizure of Yuanliu's body officially began.

Free to read.