Attributes skyrocket infinitely, I dominate the multiverse Chapter 686 Endless road, unreachable bas

After reading the memories of the three fallen world cardinal-level Xuanzhang, Mu Cang learned.

The endless Xuanzhangs in this vast territory community are not leaderless and fighting on their own, but there is a supreme commander in the real sense.

This commander who is in charge of all military affairs and power in this territory community is called "Governor of the Frontier" in the position system of the Dao Master civilization.

This title, as the name suggests, refers to the governor who guards the border area.

And this governor, in the memories of the three Xuanzhang, is exactly a genuine unattainable cardinal-level Dao Master.

At the same time, this governor is also the only unattainable cardinal-level Xuanzhang in this territory community.

And it is a strong unattainable cardinal.

Except for this governor, all other Xuanzhangs are world cardinal-level, or in other words, they are all in the huge cardinal category of world cardinal.

This is also completely understandable.

Regardless of the greatness and distance of the unreachable cardinality, we should know that within the scope of the world cardinality alone, we can completely divide into endless levels, and the gap between each level is also infinite and boundless.

So how big is this so-called "infinite and boundless"?

It can be understood in this way.

If we say that starting from the smallest infinity - ??, the road to the first world cardinality WC is so far and long.

Then starting from the first world cardinality WC, the road to the 1 world cardinality WC is also so far and long, or even farther and longer.

Why is this so?

Because the essence of the 1 world cardinality WC is that on the basis of a certain ZFC axiom system model that has introduced the first world cardinality WC axiom model, it reaches the height that can be encapsulated into a ZFC model again through various extremely complex methods.

Similarly, the path from the n-world cardinality to the n1-world cardinality, or from the x-world cardinality to the x1-world cardinality, or even from the k-world cardinality to the k1-world cardinality, is equally far and long.

These explanations and analogies are indeed somewhat difficult to understand at first glance.

So to put it more thoroughly, any large cardinality axiom actually far exceeds the proof ability or jurisdiction of the ZFC axiom system model itself.

If described in the tone of Xianxia, ​​any large cardinality is an innate chaos demon that is too powerful, so powerful that if it only relies on the ZFC axiom system's own ability, there is absolutely no possibility of any innate chaos demon being born.

Therefore, only after being settled by the chaos demon named the large cardinality, the von Neumann universe V in the "blank" state can reach a higher strength and have more colorful properties.

In fact, for countless finite, infinite, and super-finite life forms, Cantor's absolute infinity is roughly equivalent to the so-called "omniscience and omnipotence" in their cognitive scope.

However, the ZFC model, whose consistency strength is equal to or even superior to Cantor's absolute infinity, can soar to a higher level that cannot be described by the word "incredible" after having the axiom of any large cardinality.

From this, we can see how terrifying the strength of the large cardinality is. It is so terrifying that even the words such as "multiples of Cantor's absolute infinity, which far exceeds the so-called "omniscience and omnipotence" level" are not enough to describe it.

In short, when the world cardinality WC introduces the W function and then according to the ZFC substitution axiom, and then performs sup operations similar to the ? function to continuously improve the level, the universal mathematical universe that contains and accommodates the world cardinality WC will also continue to climb and upgrade together.

When this kind of promotion is truly presented in the concrete physical world, the numerical logic territory will be like a tower of Babel, while expanding the foundation in a continuous and explosive manner, it will also continue to pile up floors crazily, and the difficulty of expansion and piling up will always be so terrifying.

But this way of climbing also has its limit, and this limit is the fixed point of the world cardinality, which can also be called the "world point" of the W function.

Above this, there are also many fixed points of the world cardinality whose specific number cannot be described by the word "innumerable".

But these fixed points of the world cardinality will be blocked below by the kk? world cardinality...that is, the "great world cardinality".

No matter how huge the gap between all the world cardinality in front is, it is equally small for the great world cardinality.

Because the common tail of these world cardinality is only ω.

As for the so-called "common tail", it is an important mathematical concept in set theory, which is mainly used to describe the characteristics and precision of the unbounded subsets and sequences of well-ordered sets.

To put it simply, it is the number of terms required to reach the increasing sequence when only ordinal numbers below α can be used, so it can also be referred to as "gradient".

If we expand the concept that all world cardinals below the great world cardinal have a common tail degree of ω, that is, for all n∈N, the sequence of the smallest ∑n correct cardinal is the next basic sequence of κ with a length of ω, and for any n, ∑n1 can be used to describe a certain ∑n correct cardinal, so its strength is within the scope of the ZFC axiomatic model.

However, for the basic sequence as a whole, there is no ∑ statement that can describe all ∑n, because there is no natural number greater than all natural numbers, so this basic sequence of κ cannot be defined inside Vκ, so it cannot be used as a set to apply the substitution axiom. This basic sequence must be defined outside the ZFC model, that is, in Vκ1.

In short, above a series of world cardinality fixed points is the great world cardinality, but also above the great world--\u003e\u003e

cardinality there are endless and infinite W function fixed points, and these fixed points that are extremely far away from each other also have the same common final number.

So at this level, the common final number can also be regarded as a very rough measure of the strength between different levels.

And the higher common final number level that is "closest" to this series of all world cardinals with the common final number ω is ω1, which is equal to ??.

Above this, there are ω?, which is equal to ??, ω??, which is equal to ???, ω???, which is equal to ????, and so on. There are all kinds of common final numbers with huge differences that have no boundaries at all.

These world cardinal numbers with different common finals are usually named with various complex prefixes or suffixes.

Moreover, there will be endless, complex, infinite, terrifying and indescribable gaps between the world cardinal numbers of various levels and orders within the huge "territory" "ruled" by each common final number of each level and order.

If you want to cross this gap again and again, it will involve the mathematical concept of "unbounded closed set".

Regarding this concept, there is also a simpler pre-concept called "unbounded set".

To illustrate this concept, for example, the natural numbers in the category of ω are unbounded in ω, and because ωN, N is an unbounded non-proper subset of ω. The specific definition of the concept of "unbounded" is detailed in Chapter 677

Since there is "non-true", there must be "true".

For example, for any n∈ω, there is still n1∈ω, and there is no maximum natural number, so all positive even numbers are true unbounded subsets of ω.

This concept is relatively simple, but the concept of "unbounded closed set" above it requires more consideration... No, it is much more complicated.

Let's give an example.

For example, if c is an unbounded subset of x, and for all limit ordinals, it is altx, as long as the supremum of a is a, there is a∈c, then c can be said to be an unbounded closed subset of x.

If this paragraph is expanded, it can be considered that for the limit points taken by that series of a∈c, the result is still in c, that is, c is completely closed to the operation of taking limit points, and the sup of a series of elements in c is still retained in C.

Therefore, the properties of unbounded closed sets are like an endless filter that can continuously filter out more and more extreme "elements", but will never run out of the set range.

In short, by using many "tools" including "unbounded closed sets", along the long common mantissa "path" that runs through the entire world cardinality, you can directly reach the inaccessible cardinality.

So... what will be the common mantissa of the inaccessible cardinality?

The answer is...itself.

Yes, just like the ouroboros that bites its own tail with its mouth like a Möbius strip in mythology, which represents the supreme concepts of "eternal perfection", "infinite cycle" and "eternal existence", the first unreachable cardinality...is also an unreachable cardinality.

The first unreachable cardinality is an uncountable cardinality that can only exist after adding the corresponding unreachable cardinality axioms on the basis of the ZFC axiom system model. It is both a strong limiting cardinality and a regular cardinality.

The so-called regular cardinality refers to the cardinality whose common final digit is equal to itself.

Expressed in mathematical language, it is cfkk.

Here, cfk is the minimum length of an increasing sequence with k as the supremum.

cf can be defined on all ordinals, but regular ordinals must be cardinals.

As for the strong limiting cardinality, if expressed in mathematical language, it is...if αltk, then 2αltk, then k is the strong limiting cardinality.

Here, "" means the cardinality exponentiation, so this formula means that k cannot be reached by taking the power set of a cardinality less than k.

Similarly, ω is a strong limit cardinality, because the power set of a finite set must still be a finite set.

And because ω is also a regular cardinality, it can also be said that if the necessary condition of "uncountability" is not required, then ω is the smallest unreachable cardinality.

Think about it, compared with ω, there is actually no difference between 1 and SCG3. It can be seen that climbing up from any point in the natural numbers below will never reach ω.

So from the perspective of regularity and limit, ω is equivalent to the relationship between existence and non-existence compared to all finite numbers below ω, which is completely a conceptual fault.

Of course, ω is just a "star face" in a sense compared to unreachable cardinality.

Below the real unreachable cardinality is the unbounded multi-world cardinality level, which is far more distant and profound than imagined, and the intensity is also fault-like huge.

Many intelligent beings have a very imprecise understanding, that is, they believe that... if ω is the supremum that cannot be piled up with Arabic numerals, then the first unreachable cardinality should be the supremum that cannot be piled up with the Aleph function.

But this perception is wrong. The inaccessible cardinality is much, much larger than this perception.

If expressed in real mathematical language, that is... for the limit ordinal α, there is cf?α≤α, and because α≤?α, if?α is a weakly inaccessible cardinality, then cf?αα?α.

As for the higher strongly inaccessible cardinality, it adds a condition or requirement on the basis of the weakly inaccessible cardinality.

That is... for any cardinality λ＜k, there is 2λ＜k.

The meaning of this series of mathematical expressions is that if k is a weakly inaccessible cardinality and meets the above requirements, then it can be upgraded to a strongly inaccessible cardinality.

The "strong and weak" in this sentence refers to the meaning of the range. "weak" refers to low requirements and a wide range, and "strong" refers to a more specific and clear explanation of why the large cardinality is inaccessible.

From the definition of a strongly inaccessible cardinality, it can be known that it must be a weakly inaccessible cardinality, that is, it must be a regular and uncountable limit cardinality.

r\u003e

According to Cantor's theorem, for any cardinality λ, λ＜2λ always holds true, which means that a larger cardinality can be obtained by continuously taking power sets.

But the unreachable cardinality means that if k is a strongly unreachable cardinality, then no matter which cardinality λ＜k is smaller than it, no matter how the power set 2λ is taken, k cannot be reached.

In the end, it can be proved that if k is a strongly unreachable cardinality, then for any ordinal number α＜k, so there is 2?α＜k.

And the governor of the frontier is exactly a...master of the strong unreachable cardinality level.

"So..."

Mu Cang stroked his chin and chuckled, "How powerful is that governor?"

With curiosity, Mu Cang stepped out of the Gödel constructible universe where he was, through the hidden and complex boundary path, and stepped into the other von Neumann universe.

After his ‘living’ world cardinality axiom left, the original Gödel constructible universe suddenly fell to the level of ordinary true species and became mediocre.

On the contrary, the true species territory of the universal universe that Mu Cang had the honor to step on suddenly rose to the world cardinality level.

This magical phenomenon is the terrifying result of the heaven-defying power of [Infinite Secret Strategy].

There is no need for Mu Cang to activate the "Universal Principle" himself. This heaven-defying divine skill will automatically force the highest dominance of the territory under his feet into his control.