In fact, although they are all meaningless sources.

But now the source of the "second world" that Mu Cang is in, in terms of overall strength, far surpasses the "first world" [ultimate level]... or it can be said that

A higher-order source of Reinhardt's cardinal flow.

And the large cardinal number corresponding to the unknown heterogeneous intensity of this meaningless source is... a special-complete Reinhardt cardinal number.

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

The so-called super Reinhardt base number, as the name suggests, is a super high-level enhanced version of the Reinhardt base number.

Therefore, in essence, it also belongs to a critical point of non-trivial basic embedding, embedding itself.

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

However, since the consistency strength of this large base is far less than that of the super Reinhardt base, we will not mention it for the time being.

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

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

.

Similarly, if k is a super Reinhardt base, then γ < k will exist, so that (5γ, Vγ 1) is a model of the ZF? Reinhardt base existence axiom.

Among them, ZF? can be understood as a second-order ZF axiom system.

Yes, the ZF system has one, two, three, four, and even more orders.

In general, compared to the Reinhardt base number, the super Reinhardt base number is based on it and adds a qualification:

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

If we expand this so-called "expectation" concept in detail, it means that for all ordinal numbers a, j(k)>a must be present.

To further elaborate, the definition of super Reinhardt cardinal numbers involves the transcendence of all ordinal numbers.

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

In comparison, the Reinhardt cardinality only requires the existence of a basic embedding j: V→V such that k is the critical point of j, and does not require that j(k)>a for all ordinal numbers a, which can be super-Rhine

Hart's cardinality is exactly the opposite.

Therefore, the consistency strength of the latter is far...far better than that of the former.

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

There is absolutely no comparison.

Therefore, it is necessary to look to the higher-level "mathematical world" to find large cardinal numbers with greater consistency.

That is, A-super Reinhardt base.

Its specific definition is: for a suitable class A, if all ordinal numbers 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 base.

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

It is a greater promotion or extension of the super Reinhardt base at a higher level, so the gap between the two is so huge that it is simply indescribable.

But even so, even if it is so complex, the A-Super Reinhardt base is still far... far weaker than the Berkeley base.

Therefore, we must use it as a stepping stone, leap into endless ascension, and go to a higher level to search for a higher order and greater great base number.

That is, the complete Reinhardt base.

Regarding the definition of this large cardinality, a simplified explanation is:

If for every A∈Vk 1, there is (Vk, Vk 1) which is a model of the ZF? A-super Reinhardt cardinal existence axiom, then such k is a complete Reinhardt cardinal number.

So, can the strength of the complete Reinhardt base surpass that of the Berkeley base?

Unfortunately, still can't.

Because these two large bases cannot be clearly compared.

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

There is no way to know which of these two large bases has higher strength. We can only roughly think that the strength of the two can be marked with a slightly vague "=" sign.

So, what exactly will be the large cardinality that can truly surpass the Berkeley cardinality in terms of consistency strength?

The answer is, special-complete Reinhardt cardinality.

Or it can also be called...unbounded closed Berkeley base.

That's right, after the Reinhardt cardinal genealogy and the Berkeley cardinal genealogy rise to a very high level, there will be some mysterious blending, and then the two will become one.

This may be the magic and beauty of mathematics.

As for the so-called unbounded closed Berkeley base, which completely surpasses and surpasses the complete Reinhardt base and Berkeley base in strength, its specific definition is simply:

If k is regular and for all unbounded closed sets C?k and all transitive sets M of k∈M, there are j∈e(M) and crt(j)∈C, then such k can be called unbounded

Closed Berkeley base.

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

That is, on top of the unbounded closed Berkeley cardinality, are there even larger members of the Berkeley cardinality spectrum?

The answer is, it will exist.

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

If we expand on this, it will become a very 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 Reinhardt 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 a 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:

For all transitive sets M that satisfy 5δ 1∈M 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(M

), ultimately making: (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 seriously 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 away by Mu Cang and became one of his clones.

At the same time, Mu Cang's own life and strength level, also under the combined effect of [Infinite Secret Strategy] and [Domination Invasion], jumped from the Reinhardt base level to the infinite level without any hindrance.

level, firmly stationed at the unbounded closed Berkeley cardinality 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 sent out all the Von Neumann Universe V, Gödel of various types, strong or weak, in the "second world".

Constructable universe L, well-ordered universe WO, well-founded universe WF, despecialized 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 in charge of Taoism, those who are undecided, the natives of the great base, and the super-finite and infinite beings have all been sucked away from the sky, leaving only an empty, lonely and desolate super-class 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, not a single final fragment is left."

While 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 passed through has exceeded one million trillions.
To be continued...