Beyond the Turing Machine (II)-mysterious and infinite

Beyond the Turing Machine (II)-mysterious and infinite

Infinite infinity is mysterious.

(1) introduction to the theory of Taiji infinity:

Only two Infinity,
1. Number of Natural Numbers: A few infinity (∞ 0 ),
2. Real Number: Infinite (∞ 1 ),
∞ 1 is the ultimate big.

There are countless numbers distributed between ∞ 0 and ∞ 1. These numbers are virtual, ultra-virtual, ultra-virtual, and ultra-virtual ...... These "nothingness" are actually the numbers corresponding to the n-dimensional space.
In the future, we will call ∞ 0 the infinite sun, and ∞ 1 the infinite Yin (the infinite Moon ).
Life, thinking, intelligence, and spirit are directly related to the infinite moon.

Integers 0 and 1 can be complete, while Infinity has only two: Infinite sun and infinite moon.
Infinite Moon is the ultimate universe.

We all know that the number of real numbers is the same as the number of n-dimensional space points, and there is a one-to-one correspondence, so it is feasible to put all the points in the n-dimensional space into the real axis.

I have long felt that I was on the real axis. Now I can see that I is between ∞ 0 and ∞ 1.
A lot of numbers in a real number (such as a number π, E, or Euler's constant) are located by name. They cannot draw a specific position on the number axis, and can only be roughly located in a range. "Only in the mountains, the cloud is unknown", but it does exist. The position of the virtual number on the number axis is also the same, and I am afraid it cannot be connected to a limited range, the distribution is between ∞ 0 and ∞ 1.

We can intuitively see that the virtual number is on the number axis.
Equation x-x + 1 = 0;
X1 = (1 + SQRT (3) I)/2; x2 = (1-sqrt (3) I)/2; x1.
X = x ^ 2 + 1 continuously replace itself:
X = (... + 1) ^ 2 + 1) ^ 2 + 1

Some examples of continuous scores are also intuitive.

We can define generalized real numbers, including n-dimensional numbers.

I have proved this property of a real number: a real number does not have a number adjacent to it, any two real numbers have an infinite number ).
Set the number of adjacent to the real number N N0, then | n-n0 | should be smaller than any specified number, so according to the limit definition, N0 = N, the real number just want to black hole, suck in everything around it. Every two real numbers are infinitely far apart (any two real numbers have an infinite number). Maybe the object with the highest density in the universe is "Number" -- yes, perhaps the number is indeed an entity. (Note that real numbers do not have the concept of the next real number. In fact, they are directly related to the implications of human thinking and memory)

There are countless numbers distributed between ∞ 0 and ∞ 1. These numbers are virtual, ultra-virtual, ultra-virtual, and ultra-virtual ...... These "nothingness" are actually the numbers corresponding to the n-dimensional space. N-dimensional numbers are folded between ∞ 0 and ∞ 1.

(2) answers to the mysteries of mathematics in the past century

In the conto set theory, it is considered that there are no number of infinity. conto obtains a series of infinity, ∞ 0, ∞ 1, ∞ 2… with "The set power set base is greater than the set base ...... ∞ N ......
∞ 0 is the number of all natural numbers (Natural Numbers = rational numbers = non-exceeded numbers ......)
∞ 1 is the number of all real numbers (number of real numbers = points in a straight line = points in a line segment = points in space ......)
What is ∞ 2? Later, it was hard to think of "the number of all curves in space (which can be an n-dimensional space)", that is, the number of all possible mathematical functions (including continuous and discontinuous functions ).

∞ 3 and later one hundred have not found the actual meaning for many years and become the mystery of century mathematics.

Now I will solve this problem,
I will give the answer to this question and then prove:
∞ 3 is the number of all functions with functions as variables,
∞ 4 is the number of all functions whose values are "functions as variables,
∞ 5 is the number of all functions whose values are "functions with functions as variables,
......
This recursion goes on. In general, we can say:
∞ 3 is the number of all functions,
∞ 4 is all generic functions,
∞ 5 is the number of all generic functions,
......
∞ N is all generic ...... Number of functionics (a total of N-2 extensions ),
......
This Recursion

In fact, we need to have a profound understanding of the concept of the set's idempotence to solve this problem. From the set theory, we have learned that the Set's idempotence is a set composed of all the subsets of the set. What is an equivalent concept? I said: "The set composed of all subsets" is actually all the relationships between elements in the original set! See.

This is enough,
For example, ∞ 3 is the number of relationships between all elements of ∞ 2 (all mathematical functions). That is not the number of all functions whose functions are variable)? Understand Why ∞ 3 is the number of all functional functions.

∞ 4 is the number of relationships between all elements of ∞ 3.
∞ 5 is the number of relationships between all elements of ∞ 4.
......
Recursion

By the way, there is a simple,
∞ 2 is the number of relationships between all elements of ∞ 1 => ∞ 2 is the number of relationships between all real numbers (all possible relationships) => isn't that the number of all possible mathematical functions? (Including continuous functions and discontinuous functions, which can be n functions );
∞ 1 is the number of relationships between all elements of ∞ 0 => ∞ 1 is a real number (note that a real number can be mapped to an n-dimensional space ).

In the future, we will also discuss the consortium set theory and several non-consortium set theories.

(3) Continuation of the theory of Taiji infinity

How many C Programs are there?
How many c ++ programs are there?
How many programs are there?
......
There are only a handful of infinite answers, that is, ∞ 0.

I think of this: the number of strings composed of a limited number of characters is limited to 0 to 0 (of course, the types of characters here are limited ).
We all know that 2 ^ ∞ 0 = ∞ 1 (there can also be a finite value of N ^ ∞ 0 = ∞ 1, n is> 1). For programs on computers, its character types are limited. If we regard all programs as strings, we can know that the answer is ∞ 0.

Or simply regard the source code of the program as the 01 sequence, and the length of this sequence is limited.

How many rational numbers are there?
Consider the rational number definition, which is an infinite loop decimal number, and the length of the loop section must be limited. Therefore, according to "The maximum number of strings composed of a limited number of characters is ∞ 0 ", the number of cyclic nodes is limited to ∞ 0, and the integer part is also limited to ∞ 0, while ∞ 0 multiplied by ∞ 0 = ∞ 0, so there are ∞ 0 rational numbers. This conclusion is also found in mathematics textbooks.

How many algebra are there?
The algebraic equations are finite and long, so there are ∞ 0 Algebraic Equations in total-you can understand it.
The root number of the algebraic equations is ∞,
∞ 0 multiplication ∞ 0 = ∞ 0, so the number of algebra is ∞ 0. This conclusion is also found in mathematics textbooks.
A non-algebraic number is beyond the number. There are between ∞ 1-∞ 0 = ∞ 1 beyond the number of algebra.
However, beyond the number, humans still have limited knowledge, such as π, E, and Euler's constants ......

We need to pay attention to constants like π, E, and Euler ...... This part of the beyond is regular, with various series expansion, continuous score expansion, and root expansion ...... And so on, then our current problem is:

How many regular numbers are there?
Note that this rule is arbitrary as long as there is a rule. Can you think of the answer? The answer is ∞ 0.
Because the pattern can be described with a finite string, and the type of the character element is limited, or there are only two, because it can be encoded into a series of 01.
To put it bluntly, the number of characters that can be described with a finite length is ∞.
How many irregular numbers are there? There are ∞ 1-∞ 0 = ∞ 1, far exceeding the regular number.

When I started learning advanced mathematics that year, how wonderful I saw the expansion of π series! I thought we could find out the rule in this way, and then I realized that it was far from enough!

How many people can describe the number?
∞ 0, because the description must use a finite length string.

How many people can know?
∞ 0, because what a person knows must be a finite string.

Although I still cannot write a "number that people cannot know", I know that there are more than one number that people can know, no matter how many people know each other.

Godel's Incompleteness Theorem points out the limits of the capabilities of the logical system, and now raises the following question:
What are the total problems that a logical system can solve? ∞ 0
What can't be solved? ∞
It is still because the length of the string is limited. From this perspective, we can easily understand Godel's incompleteness theorem.
The incompleteness of Godel originates from the limit of ∞ 0. In addition, we can see that the majority of unsolved problems exist.

Next, let's look at the total number of cosmic patterns that people can recognize?
∞ 0? Because we must be able to write a finite-length string to be recognized ...... (Unless ...... Maybe ......?)

Oh, my God, there is a law that no one knows (unless ...... Maybe ......?), However, in the 0-0 rule that can be recognized here, even these words are included, will there be any other breakthroughs?

By the way, how many cosmic patterns are there in total?
Think for yourself ......

I was scared when I thought about what I wrote today. Do readers feel scared here? How can this problem be solved.

Now let's think about this:
It is proved that at least ∞ 0 problems can only be solved by the reverse verification method.
Next:
The answer to the question raised with a limited character is not necessarily resolved using a finite long string. (If any, change to none)
Also:
Any question can be solved, but the answer may be infinitely long. Sometimes the reverse verification method is the only solution.
Also:
"What is the nth digit of any number R (such as π, E) in the M-base system? Think about what it means to realize it?

(4) multiplication and idempotence

As I have already said, the idempotent set (a set composed of all subsets) is actually all the relationships between elements in the original set. For example, the power set of a real number is all mathematical functions.
We can get another equivalent concept of idempotence: idempotence controls all the laws of the original set transformation.
The final set must achieve this: all the rules that control the transformation of this set are the elements of this set, all in this set.
This collection is called the "Universe" and "a collection of all sets and elements ".

The power set is the "consciousness" of the original set. 0 ^ 0 = 1. The essence of multiplication is also "consciousness ".
Both the Power Set and the multiplication operator sublimate the "input.

The theory of Taiji Infinity has the following formula:
Base C ≦ P (C), C is a set, and P is a power set. The equal sign is valid only when this set is an integer.
So there are only two Infinity values: ∞ 0 and ∞ 1,

The difference between this formula and the set theory of modern mathematics is that in the equal sign, the set theory of modern mathematics is C <p (c ).

Modern Mathematical set theory has a very misleading idea. It is just a matter of practice to avoid all kinds of paradox.
For example, in order to avoid the paradox of U <p (u) in the consortium set theory, the zfc system does not have a rigid definition of "including all sets, this means that the universe does not exist with your overbearing eyes wide open. Can you believe it? I don't believe it. In addition, the NBG system retains the "set of all sets ".

First, I copied the conto process to prove C <p (c) as follows, and then analyzed the error:
The following is a copy of a popular version from the Internet. You can open a Collection Theory Textbook, which proves it.
---------------------------------
Definition: a set of all subsets of set a is called the power set of,
P [A].
This definition seems very simple. For example, if a = {a, B, c}, A has
Therefore, the Power Set of A is the set that contains the eight subsets, namely:
P [a] ={{}, {A}, {B}, {c}, {a, B}, {A, c}, {B, c }, {a, B, c }}
Please note that the empty set {} and set a are two elements of the Power Set of A. No matter what set a we use, this is correct. We should also note that the power set itself is also a set. This basic fact is sometimes overlooked, but it is of great significance in Kanto's thinking.
Obviously, in the above example, the base of the Power Set is greater than the base of the set itself. That is to say, set a contains three elements, and its idempotent set contains 23 = 8 elements. It is not hard to prove that a set containing four elements has 24 = 16 subsets, And a set containing five elements has 25 = 32 subsets. In short, A set of n elements a has 2n subsets.

But what if A is an infinite set? Is the base of the Power Set of an infinite set greater than that of the set itself? Conway's theorem answers this controversial question:
To prove this, we must follow the definition of the strict inequality between the over-limit bases in Kanto described earlier in this chapter. Obviously, we can easily find a one-to-one correspondence between A and some P [a], because if a = {A, B, C, D, E, ......}, We can make element a correspond to the subset {A}, so that element B Corresponds to the subset {B}, and so on. Of course, these subsets {A}, {B}, {c }...... It's just a deficiency in all the subsets of.
So far, everything is simple. However, it is necessary to prove that a and P [a] do not have the same base. We use the indirect proof method, first assuming that their bases are the same, and then export logical conflicts from them. That is, we assume that there is a one-to-one correspondence between all of a and all of p [. To facilitate the demonstration, we will use an example consistent with this assumption for further reference:
Therefore, this arrangement indicates that there is a hypothetical one-to-one correspondence between all elements of a and all elements of P [. Note that some elements of a belong to their subsets in this ing relationship. For example, C is the corresponding set {A, B, C, d} element. On the other hand, some elements of a do not belong to their corresponding subsets. For example, a is not an element of its corresponding subset {B, c.
What is incredible is that this split of incompatible things provides clues that lead to this proof of logical conflict, as we can now define Set B as follows:
B is the set of elements in the original set a that do not belong to its corresponding subset.
Based on the corresponding relationship of the above assumptions, we can see that a and B (because B is not an element of {d}), D (D is not an empty set of course), and g (not {H, i, j ,......} All belong to Set B. However, C, E, and F are not elements of B, because they belong to {a, B, c, d}, A, and {A, C, F, g ......}.
Therefore, Set B = {a, B, d, g ,......}. In this way, B is a subset of the original set. Therefore, B belongs to the Power Set of A, and it is bound to appear in a position in the right column of the corresponding relationship. However, based on the one-to-one correspondence originally assumed, we also conclude that there must be a certain element y of A in the left column corresponding to B:
So far, everything went well. But now we have a fatal question: "Is y the element of B ?" Of course, there are two possibilities:
In the first case, assume that Y is not an element of B.
According to our initial definition of B, In the original set a, each set of elements that do not belong to its corresponding subset ", we can see that Y is a member of B, because in this case, Y is not the element of the corresponding set.
In other words, if we first assume that y does not belong to B, we are forced to conclude that y should be the element of B. Obviously, this is self-contradictory, So we exclude the first case, because this is impossible.
In the second case, assume that Y is the element of B.
We turn to B's definition again. In the second case, if y belongs to B, y should conform to B's definition, that is, Y is not the element of its corresponding set. Sorry! The set corresponding to Y is exactly B. Therefore, y cannot be the element of B.
In this case, given that Y belongs to B, we have to conclude that Y is not an element of B. As a logical structure, we once again entered a dead end.
Something is wrong. The first and second cases are the only two possibilities, but both may cause logical conflicts. We conclude that there must be an incorrect assumption somewhere in the argument. Of course, at the beginning, we assume that there is a one-to-one correspondence between A and P [. Our paradox clearly destroys this assumption: This correspondence cannot exist.
----------------------------------------------

This proves that there are two very hidden errors:
1. In this process, set a is a set of only a few by default (the set of elements that can be written one by one must be a few ).
2. The reasoning based on the set theory is unreliable, and the Russell paradox is not so well resolved.

At the beginning, I tried to prove that "an unmeasurable set cannot be idempotent", but later I found that "an unmeasurable set is still its own ".
So there are only two Infinity values: ∞ 0 and ∞ 1,
What does this mean? Addition, subtraction, multiplication, division, and other operations are real numbers! -- I have long guessed that all these things are numbers, sin, cos, Lim, cosine, Σ, cosine ...... All are numbers, not only in the encoding sense, but in the real sense, and can be indexed by name on the Real Axis!

The laws of real numbers are mapped to real numbers, so real numbers are "self-controlled", and like life-consciousness can realize itself.

Just now, I think that "the power set controls all the rules of the original set transformation" can also be said that "the power set is defined as all the operations and input and output on the original set ".

From the perspective of String Length, we can still prove that the power set of an unmeasurable set is itself.
Each real number can be expressed as a 01 sequence with a length of ∞ 0. Therefore, there are 2 ^ ∞ 0 = ∞ 1 real number,
Each function can be encoded and expressed as a 01 series with a length of ∞ 0. Therefore, there are 2 ^ ∞ 0 = ∞ 1 functions-and the previous function was ∞ 2, so ∞ 1 = ∞ 2.
Function generic function ...... So:
∞ 1 = ∞ 2 = ∞ 3 = ∞ 4 = ∞ 5 = .........

 

I even found that the concept of designing a set is unreasonable. In particular, the concept of a set and an element must be unified in an infinite state, (this may be the only way to solve the Russell paradox-not to distinguish between sets and elements ). I used this idea when designing the Taiji language. Functions, classes, and variables are described in the same data structure. In this way, variables have all the features of classes, for example, you can overload a class or a specific variable. The design concept is "a set is an element, and an element is a set ".

Note that the number and order of sets are two concepts.
Real numbers cannot be counted, but can be sorted. Is to sort by the digital sequence.
For example, functional analysis first introduced a "good order Theorem", saying that "each set can assign a sequence and make it a good order set ", it can be seen that each unmeasurable set is similar to a real number which can be encoded into character sequences for sorting. Here, we can also get ∞ 1 = ∞ 2 = ∞ 3 = ∞ 4 = ∞ 5 = ..........

(To be continued)