In the mathematical theoretical system [transfinite ordinal numbers], there are so-called three types of conditions.

1. Anti-reflexive:

That is, if a≤b, and b≤a, then a=b.

2. Transitivity:

That is, if a≤b, and b≤c, then a≤c.

3. Completeness:

If a≤b or b≤a, then there is no incomparable situation.

In fact, ‘≤’ in the category of natural numbers to real numbers known to all intelligent beings conforms to these properties.

These properties are the basis for the [total order relationship] between various types of collections.

As for the so-called total order relationship, it is a comparison operation at the set level. (See Chapter 580 for details)

If a one-to-one correspondence can be established between any two well-ordered sets.

Then, it can be said that they are [same ordinal numbers].

In fact, it is not just ordinal numbers. In the vast field of mathematics, there are also a large number of definitions that establish the similarity of the properties of two objects through some kind of one-to-one corresponding transformation.

Its name is also quite similar to the concept of "same ordinal numbers".

For example, isomorphism, homomorphism, etc.

If we want to give a more detailed and vivid metaphorical description of the concept of "same ordinal numbers", then we can use the realm of "Galactic Overlord" as an example.

In the realm of Galactic Overlord, if you base your strength on the level, start from the lowest level and count all the way up.

Second level, third level, fourth level...all the way up to the top tenth level top overlord.

Then this power leveling system has a total of ten levels.

According to their strength, they form a well-ordered set from small to large. (For details on the definition of a well-ordered set, see Chapter 580)

At the same time, the natural numbers from 1 to 10 can also form a well-ordered set.

Obviously, there is a one-to-one correspondence between Galaxy Overlord levels 1 to 10 and natural numbers 1 to 10.

And the corresponding structures of the two also maintain the order.

Therefore, it can be said that the [Galaxy Overlord] hierarchy and the set of natural numbers 1 to 10 are [same ordinal numbers].

It can also be said more simply that the ordinal number is 10.

Extrapolating this to a larger level, then all natural numbers can obviously form a total ordered set, or a well-ordered set.

However, it is not a finite set, but an infinite set.

This infinite set is the smallest transfinite ordinal number w, and it is also Mu Cang's level of strength when he first reaches infinity.

Of course, this is only the level at which He first ascended to the infinite time.

As for the current Mu Cang, he is already far above the W level.

But w... is already truly infinite.

How can we transcend infinity?

The answer is, it can be surpassed.

However, you need to open your mind and start a thinking storm.

start!

Question, how to obtain a higher-order and larger transfinite ordinal number by adding an element to the natural number set w?

At first glance, this seems impossible.

Because there are already infinite elements in the natural number set w.

If you want to add another element while maintaining the properties of w as a well-ordered set, where should you add it?

Without thinking about the answer first, you can flip the question around.

After flipping it is... Can we take enough elements from all natural numbers w to construct a smaller infinite ordinal number?

As long as you think about it for a moment, you will know that this problem is very similar to the "Hilbert Hotel Paradox Problem", or in other words, they are very different. They are both thinking and discussing infinite sets.

In short, even if any number of elements are removed from the set of all natural numbers w, as long as there are infinite elements left, w will still have the same ordinal number as all the natural numbers.

Now that the problem has been flipped, let’s flip the conclusion again.

After flipping it, it is meaningless to add any number of elements to w.

Even if it is added, the result is still a set of ordinal numbers of the same size as the set of natural numbers.

So, what should we do now?

What can we do to break through w and reach the higher level of infinity?

It's very simple, add an element to the [end] of all natural numbers.

However, there are infinitely many natural numbers in total. How can we add an element to the so-called "end" that is impossible to exist according to common sense?

Note that this is the key point in the theory of [transfinite ordinal numbers].

Very important!

If you can understand this key point, you can understand how to add an element to the end of all natural numbers.

Then it will be very easy, it can even be said to be a matter of course, to completely understand Mu Cang's current level of strength.

But if you can't understand.

Then, let’s treat Mu Cang as a general infinity.

Because for all finite creatures, no matter which level of infinity there is, there is not much difference; they are all levels of God that can never be reached.

Now, let’s start thinking.

Let’s think about it first, why should we add an element to the [end] of all natural numbers?

The reason is that we want to get a transfinite ordinal number larger than w, and then get closer to understand the level where Mu Cang is.

According to the definition in ordinal number theory, ordinal numbers must be a well-ordered set that can be sorted sequentially.

So if you want to 'expand' a series of all natural numbers that have been arranged, of course you can only add elements at the [end].

However, according to the original size ratio method for all natural numbers w, it is obviously impossible to find any number that is larger than all natural numbers.

Therefore, it is necessary to slightly modify the definition of "order relationship" in ordinal theory, and then find another method of comparing sizes, so that the exploration of breaking through w can continue.

So I kept exploring like this, and kept exploring.

Finally, it can be found that in the [set theory] system, there is a natural method of comparing sizes.

That is, it is a [subset], or it can be called an [include] relationship.

From this, you can try to redefine the natural numbers by using the [set] method.

This chapter is not over, please click on the next page to continue reading! It should be noted that this method was used in many human civilizations on earth in the three-dimensional universe by the father of game theory and the father of computers - John von Noy

Founded by Mann.

Let’s start with:

Because the smallest set is the empty set, then 0 can be defined as the empty set.

That is: 0=?

Then for 1, it can be naturally defined as a set with one element.

This element is 0.

That is: 1={?}={0}

Continuing, for 2, it can also be defined as:

2={0,1}

For 3, it can be defined as:

3={0,1,2}

From this, the analogy continues.

Then, we can finally deduce that all natural numbers N are the sets with a total of n elements from 0 to n-1.

That is: N={0,1,2,3……n-1}

Even if all natural numbers are redefined and combined with the [subset] relationship, they will still be a well-ordered set.

Because it meets various conditions of [Ordinal Number Theory].

After reaching this step, you can consider adding another element to the [end] of the entire set of natural numbers.

Then...wait a minute!

Have you discovered a rule regarding the construction of natural numbers?

That is, after each natural number is constructed, it actually adds the previous natural number [self] as an element to the set of its [self].

Think about it, 1, 2, 3, 4... are they all like this?

Yes, indeed.

So, now if you add the entire set of natural numbers itself as an element to the set of natural numbers, what will you get?

Give it a try.

Many times, people habitually record the set of natural numbers as N.

However, in the ordinal number theory system, the entire set of natural numbers is usually recorded as w.

Therefore, w can ={0,1,2,3...n}

Then, if w is added to its own set, it is: {0,1,2,3...n...w}

So, is this set well ordered?

Yes, it is a well-ordered set and the real thing.

Because any two elements among them can be compared in size.

And w contains all other elements, and all other elements are also subsets of w.

Therefore, w should be ranked last when sorting.

There is no doubt about it.

In short, the operation of "adding an element to the end of all natural numbers" was finally successful.

The breakthrough for w was finally successful.

The new transfinite ordinal number obtained through this operation is the previous {0,1,2,3...n-1...w}.

That is, w 1.

Note that the 1 here is not the addition of a natural number 1, that is a completely different matter.

At the same time, w cannot simply use the four arithmetic operations of addition, subtraction, multiplication and division. That is a big mistake.

Because the sum of set ordinal numbers is defined after defining a certain well-ordered relationship on the non-intersecting union of two well-ordered sets.

In addition, after obtaining w 1, a larger transfinite ordinal number that cannot establish a one-to-one correspondence with the set of natural numbers.

Then we can get w 2 by repeating the previous operation of adding w to itself to get w 1.

Then add w 2 to itself to get w 3.

By repeating this operation, you can get w 4, w 5, w 6, w 7...

By analogy, after performing an infinite number of such operations, we can reach the limit of this infinite and infinite road - w w.

That is, w·2.

w can be called the first level of infinity, and w·2 can be called the second level of infinity.

In a sense, the gap between the two cannot be described even by the simple word "infinite".

Also note that w·2≠2xw.

w·2 is equal to w w, which is also equal to wx2.

In other words, 2xw≠wx2.

These two are completely different ‘things’.

The latter is a higher-order infinite ordinal number that is much, much, much larger than the natural number set w.

After reaching this level, similar to the previous 'addition' situation, there is no commutative law of multiplication among ordinal numbers.

If we simply interpret w·2 as 2xw, it would also be a big mistake.

Because 2 2 2... keep adding, adding w times in total, the final result will still be w.

In short, after getting w·2, you can continue to use the previous method, infinitely and infinitely to get w·2 1, w·2 2, w·2 3, w·2 4...

By analogy, we finally get w·2 w.

That is, w·3.

Now that we have w·3, we can naturally obtain w·4, w·5, w·6 by replicating the previous infinite and infinite methods... and finally get wxw, which is w2.

This is another higher-order transfinite ordinal number that far surpasses all the previous transfinite ordinal numbers.

At the same time, since we have w2, we can naturally obtain w3, w?, w?, w?... by continuously performing various more complex operations than before.

It finally breaks through to w?, which is w^w, and can also be written as w↑w.

And Mu Cang, and the boundless ocean of time beneath his feet, are exactly in two deep levels that are not too far above the huge transfinite ordinal number w↑w.

It's just that Shi Wangyang's level is lower, and Mu Cang's level is higher.

In short, the gap between the two is not particularly 'big' on the whole.

This is exactly why Mu Cang was able to trample on the endless past and future of the boundless sea so wantonly.

Because in the realm of real infinity, even if there is only a slight difference, it will inevitably be an infinite gap.

In addition, Mu Cang also clearly knows that the power of chaos is not just that.

〖w↑wAbove〗 is definitely not the end of his surge in strength.

There was some kind of power that suppressed Mu Cang and suppressed the chaos of chaos.

"The source of this oppressive power...the Arrow of Meaning."

Mu Cang slowly raised his head and looked at the... huge bronze arrow that existed in the endless distance above the ocean of time.

That's right, it was the mysterious bronze arrow that hung over the endless past and future of the boundless sea - the Arrow of Meaning, that suppressed his growth in strength.

not only that.

Mu Cang could clearly feel that the endless ocean of time below was severely suppressed by this bronze arrow of unknown origin.

If it were not for the suppression of the Arrow of Meaning, the magnitude level of Time Vast Ocean would most likely be elevated by several levels.

But for this arrow of meaning that exuded such terrifying and suppressive power, Mu Cang understood from somewhere that maybe... it didn't deliberately suppress anything or restrain anything at all.

This arrow may still be in some very deep state of 'sleep'.

In other words, the terrifying suppression that Mu Cang and Shi Wangyang endured was most likely just a little pressure naturally exuded by the Arrow of Meaning.

In other words, this mysterious arrow of meaning simply hung there, severely suppressing Mu Cang and the ocean of time.

As a result, Mu Cang was unable to be promoted to the level he was supposed to be promoted to.

It also made the ocean of time fall hard from the level where it should have stood.

However, Mu Cang also has a solution to this problem.

The method is to leave this vast ocean of time.

The simplest way to escape is to complete the ninth realm of Haoyang Method, condense the infinite rank points, break away from the phenomenal world, and step into the essential world.


This chapter has been completed!