Attributes are rising infinitely, I dominate the multiverse Chapter 693: Upgrading three levels in a

After completely engulfing all of the Undetermined General's [Free Circuit], and then possessing the bizarre and abnormal power of the [Abyss Magic Number] level.

Mu Cang suddenly "took off" the vest of the Undetermined, and began to let his body face the abnormal power that was far beyond the completely indescribable cardinal level.

And under the stimulation of this high-level abnormal power, [The Ultimate Secret Strategy] was also activated instantly without any hindrance.

Buzz——

So in just a moment, Mu Cang's foundation, body, profound marrow... everything, suddenly soared from the completely indescribable cardinal level to the more terrifying and magnificent measurable cardinal level.

Boom boom boom——

Instantly, the entire army of the undetermined measurable base order of magnitude, which was originally led by the [Free Circuit] in the vast and narrow abyss, and was originally a huge and magnificent army, which was built by the endless and boundless and bizarre territories as soldiers, suddenly froze, and then fell apart and completely disintegrated.

This was just the result of Mu Cang's expansion of a little power.

It can only be said that the huge and majestic axiom structure of high-order large cardinality such as the measurable cardinality is far from being as simple as being one, two, three or four levels higher than the completely indescribable cardinality.

If you want to understand it even in the simplest way, you have to go through an extremely complicated and long journey.

For example, the indescribable cardinality refers to the large cardinality defined by the concept and model theory tools of the ∏n or ∑n formula.

If its detailed mathematical structure is unfolded, it is: if for any ∏n formula ΦX containing only one second-order free variable X, there is an α-layer structure 〈Vα, ∈?Vα, R〉 that satisfies ΦR, that is, 〈Vα, ∈?Vα, R〉?ΦR holds, then there exists βltα, and the β-layer substructure also satisfies ΦR, that is, 〈Vβ, ∈?Vβ, R∩Vβ〉?ΦR∩Vβ, then the cardinality α can be called the ∏n indescribable cardinality.

As for the reflection principle entangled in this complex mathematical structure, it means that any first-order formula in the whole domain can be replaced by a relativized formula in a certain layer Vβ.

In addition, the so-called indescribability can also be understood as a formula that can be "true" in the α-layer structure must also be "true" in a certain β-layer before α.

From this, we can deduce many conclusions or produce many results, such as... If κ is a strongly inaccessible cardinality, then κ is a ∏10 indescribable cardinality if and only if κ is a ∑11 indescribable cardinality, and if and only if κ is a weakly compact cardinality, then κ is a ∏11 indescribable cardinality if and only if κ is a ∏11 indescribable cardinality.

Above this, there is a strongly expandable cardinality.

In logical form, if a cardinality κ is λ-indecipherable if and only if for each negative power set of the transitive model M of the cardinality κ of ZFC, and κ is in M ​​and M contains all its sequences of length less than κ, then the elements in M ​​are embedded into the transitive model according to the relation j non-trivial basis, where the critical point of j is κ and jκ≥λ.

If a cardinality is expandable, then it is λ-indecipherable for all ordinals λ.

At the same time, if a cardinality κ is strongly λ-unfoldable if and only if for each ZFC negative power set the transfer model M of cardinality κ is such that κ is in M ​​and M contains all its sequences of length less than κ, then there exists a non-trivial basic embedding j of M into the transfer model "N" where the critical point of j is κ, jκ≥λ, and Vλ is a subset of N.

Finally, since the last N contains all its sequences of length λ, if a cardinality is strongly expandable, then it is strongly λ-unexpandable for all λ.

These properties of strongly expandable cardinality are actually weaker versions of supercompact cardinality, consistent with VL, so the existence of strong expansion also implies the consistency of the weaker version of the appropriate forcing axiom.

And standing on this large cardinality is the Ramsey cardinality, which and its theorem establish the specific properties of ω with R cardinality generalized to the uncountable case.

That is, if κltω denotes the set of all finite subsets of κ, then an uncountable cardinality κ is called R, and if for every function fκltω→0,1, there is a set A of cardinality κ that is homogeneous with respect to f.

Expanding on this, for every n, the function f is constant on subsets of cardinality n from A, and if A can be chosen to be a stationary subset of κ, then the cardinality κ cannot be called R.

If for every function, the cardinality κ is actually called Rfκltω→0,1, and C is a closed and unbounded subset of κ, then for every λ in C there is uncountable cofinality, and there is an unbounded subset of λ that is homogeneous with respect to f.

Ignoring all the various complex mathematical structures, the upper limit of all strong Ramsey cardinality is a measurable cardinality.

This large cardinality is uncountable κ, and there is an additive, nontrivial, 01-valued measure on the power set of κ.

At the same time, κ is measurable, which means that it embeds the non-trivial basis of universe V into the critical point of the transmission class M.

In short, the measurable cardinality is a kind of possession, or the use of the high-order cardinality of the super-strong structure in the model theory.

After possessing such a shocking power, Mu Cang was not satisfied. Instead, he immediately put on the identity vest of [Free Circuit] and sent an insulting message to the higher-level general who governed [Free Circuit] and its legions through [Juewang Nest Domain]:

"Old beast, you are so ugly, why are you still hanging around in the world? Why don't you hang yourself to death!"

However, after the "text message" was sent, not long after, only one message with the content "?!" was sent to Mu Cang through [Jue Wang Nest Domain].

"Huh? You have such a good temper, it's rare."

Mu Cang's eyes flashed, and then he edited another message that was much more spicy than the previous one--\u003e\u003e

and sent it over.

Buzz——

Sure enough, this message finally detonated the emotions of the other party's undecided general, causing him to have a surge of malice towards Mu Cang.

And under the stimulation of this extremely distant malice, [Wu Jue Secret Strategy] was also activated instantly, and Mu Cang's original body was instantly teleported across domains to a huge and bizarre territory with a scale and strength equivalent to Wood's base. Wood cardinality can be read as Wuding cardinality

Of course, in the anomalous number system owned by the Undetermined, the abnormal number that can be strongly matched with Wuding cardinality should be called... [Xuangou number].

As for the so-called Wuding cardinality, which is deeply related to the determinism of real number sets, the internal model theory, and the absolute theory of set theory universe, it can be considered as a strong limit point that completely transcends various unreachable cardinality classes, and there is no other strong limit point below it.

At the same time, the axiomatic structure of Wuding cardinality is also one of the core components of the G?del-type construction of provable set universe (L).

The mathematical construction of this high-order large cardinality is: if the cardinality δ is a Wudin cardinality, if for each function fδ→δ there exists κ＜δ and f“κ?κ, and there exists an elementary embedding jV→M with a critical point κ, then Vjfκ?M.

Therefore, assuming that δ is a Wudin cardinality, and G?P＜δ is a V-type general filter, then there exists an elementary embedding jV→M, M transfer in VG, so that VG2M＜δ?M and jδδ.

In addition, for all a∈P＜δ, a∈Giffj“?a∈j.

In addition, although the large cardinality axioms can be compared by comparing the [consistency strength], a relatively clear linear hierarchy that can be used to compare which one is larger and which one is smaller can be obtained.

But this does not mean that these large cardinality axioms will show an absolute disconnection relationship.

On the contrary, the connection between these large cardinality is much closer than imagined, and much closer.

In short, the world of large cardinality is very complex, abstract and profound, and cannot be completely matched with the physical world.

After possessing the abnormal power of [Xuangou Xishu] level, Mu Cang routinely "took off" the vest of the undetermined person.

He began to let the measurable cardinality level essence, mysterious marrow, and foundation of his own body face the stimulating threat from [Xuangou Xishu].

Immediately, [Wu Jue Mi Ce], which could not bear the stimulation, was immediately activated, and without any process, everything of Mu Cang was suddenly upgraded from the measurable cardinality to the Wu Ding cardinality level.

But strangely, after completely transforming and upgrading to a perfect Wu Ding cardinality level life form, Mu Cang did not immediately "touch porcelain" the next "lucky person", but instead shifted his attention to the memory of the undetermined general just now.

Because He discovered a big secret in the other party's memory.

That is, the birth of the meaningless source is likely to be closely related to the core ancestral land of the Dao Master Clan - the All-Knowing Tower.

And one of the most important reasons why the Undecided Clan established the Free Kingdom and continuously gathered military forces to attack the Dao Master Clan was to be able to reach the All-Knowing Tower in the center of the Inevitable Kingdom and climb to the top of the tower.

"The All-Knowing Tower... So that's it. It turns out that the purpose of the Undecided to launch a war is to climb the tower."

Mu Cang concentrated on thinking, "And the reason why those Dao Masters established huge main and branch defense lines is to defend the tower. What a large-base version of King of Glory."

Then, He pondered again:

"So, according to this intelligence, could the emergence of the Undecided Clan... be related to the [True King of the Tower] who ruled all Dao Masters? Could it be that He... contributed to the birth of countless Undecided? But if so, what good would it do to Him?"

Obviously, Mu Cang couldn't think of an answer to this question.

It seems that only by going up to the All-Knowing Tower and asking the True King in the future can we know.

However, in the memory of the Undecided General, there is a message saying that if you want to climb the All-Knowing Tower, there are conditions.

But what exactly is this so-called condition, the Undecided General does not know.

"Forget it."

Mu Cang shook his head slightly, "As long as I am strong enough, I will definitely know everything I should know."

After that, He started another round of "touching porcelain", and the target was the Undecided Marshal named [瀷灞].

According to the information, the Marshal who led the infinite high-level Undecided Legion, his life level is at the so-called [fight?昡shu] level.

And the large base number of strength corresponding to the [多?瀬肖] is... the super-compact base number.

Regarding this large cardinality, if described in mathematical language, it is... For any cardinality κ, a supercompact cardinality λ satisfies that for all κ less than λ, there exists a κ-compact ultrafilter.

Through this κ-compact ultrafilter, the supercompact cardinality κ can "capture" or "control" the structure of all sets smaller than κ.

More specifically, for any cardinality κ less than λ, there exists an ultrafilter U such that for any set A less than λ, if the cardinality of A is greater than or equal to κ, then A belongs to U.

In addition, for any family of sets with less than κ, if the intersection of each set in the family is non-empty, then the intersection of the entire family is also non-empty.

Finally, if a cardinality κ is ultracompact, if for every regular cardinality λ less than κ there exists an ultrafilter U such that U is not only an ultrafilter, but also for any function from λ to κ there exists a set in U such that the image of all elements in the set under the function is smaller than themselves.

As for the so-called κ-compact ultrafilter, it can be considered as a family of sets that satisfies the definition of an ultrafilter, and the cardinality of each set in this family is at least κ, and this ultrafilter has κ-compactness for families with less than κ sets.

So what is an ultrafilter--\u003e\u003e

?

Its specific definition is that it is a special set family with non-emptiness, ultrafilter is not empty; closedness, if sets A and B are in the ultrafilter, then their intersection A∩B is also in the ultrafilter; superset property, if set A is in the ultrafilter, then any superset of A is also in the ultrafilter; and finite intersection property, the intersection of any finite sets in the ultrafilter is still in the ultrafilter.

Therefore, the ultra-compact cardinality is a higher-order cardinality far superior to the Wu Ding cardinality.

Skipping the process of "touching" the [瀷瀞] Marshal, in short, after [Wu Jue Mi Ce] started running again, Mu Cang once again teleported across domains and descended into an extremely vast and vast huge and bizarre universe.

And in the moment of descending here, Mu Cang also very smoothly swallowed and merged all the essence, power and memory of the [瀷瀞] Marshal, reaching the [ Dou·Xuanshu] level.

Afterwards, Mu Cang "took off" his identity vest again, and once again let his essence, foundation and profound marrow face his strange and bizarre power. The stimulating [Infinite Secret Strategy] was activated again, and everything he had suddenly rose to the super-tight base level.

But at this moment, Mu Cang suddenly smelled the breath of the final fragment from the depths of [Ling Ba]'s huge memory...