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?
To be continued...