Hilbert’s First Problem and the New Progress of Infinity Theory

: In the 19th century, Cantor created the infinite cardinal number theory based on the "1-1 correspondence" principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of "whole is greater than part", and created another ruler for measuring infinite sets. At the same time, the development of the infinity theory provides new ideas for solving Hilbert's first problem, and provides a new mathematical foundation for Cosmic Continuum Theory. Abstract: In the 19th century, Cantor created the infinite cardinal number theory based on the "1-1 correspondence" principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of "whole is greater than part", and created another ruler for measuring infinite sets. The development of the infinity theory provides new ideas for solving Hilbert's first problem, and provides a new mathematical foundation for Cosmic Continuum Theory.


Introduction
In 1874, Cantor introduced the concept of cardinal numbers based on the "1-1 correspondence" principle. Cantor  In 1938 Gödel proved that the CH is not contradictory to the ZFC axiom system. In 1963, Cohen proved that the CH and the ZFC axiom system are independent of each other. Therefore, the CH cannot be proved in the ZFC axiom system [2]- [3].
However, people always have doubts about infinity theory. For example, in the study of Cosmic Continuum, the existing infinity theory shows great limitations [4]- [14].
In the 21st century, Sergeyev started from "the whole is greater than the part" and introduced a new method of counting infinity and infinitesimals, called the Grossone method. The introduced methodology (that is not related to non-standard analysis) gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. [15]- [40] The Grossone method introduced by Sergeyev takes the number of elements in the natural number set as a total number, marked as ①, as the basic numeral symbol for expressing infinity and 2 infinitesimal, in order to more accurately describe infinity and infinitesimal.
The Grossone method was originally proposed as a Computational Mathematics, but its significance has far exceeded the category of Computational Mathematics. In particular, the Grossone method provides a new mathematical tool for the Cosmic Continuum Theory. A new infinity theory is about to emerge.

The traditional infinity paradox and the fourth mathematics crisis
In the history of mathematics, there have been three mathematics crises, each of which involves the foundation of mathematics. The first time was the discovery of irrational numbers, the second time was the infinitesimal problem, and the third time was the set theory paradox [41]- [42]. However, no one dare to say that the building of the mathematical theory system has been completed, and maybe the fourth mathematical crisis will appear someday.
In fact, the fourth mathematics crisis is already on the way. This is the infinity problem. In 1900, Hilbert put the Cantor continuum hypothesis as the first question in his famous lecture on 23 mathematics problems [43]. This will never be an impromptu work by an almighty mathematician.
The infinitesimal question unfold around whether the infinitesimal is zero or not. From the 1920s to the 1970s, this problem has been initially solved through the efforts of generations of mathematicians.
However, there are still different opinions about the second mathematics crisis. I believe that the infinitesimal problem has not been completely solved, otherwise there would be no infinity problem.
Because the infinity problem and the infinitesimal problem are actually two aspects of the same problem.
Let us first look at what is problem with infinity.
The first is the expression of infinity. Now, there are two ways to express the infinity, one is to express with infinity symbol  , and the other is to express with infinity cardinal number. However, neither the infinity symbol  nor the infinity cardinal number can effectively express infinity and infinitesimal.
For example: when expressed in the infinity symbol  , we cannot distinguish the size of the natural number set and the real number set, nor can we distinguish the size of the natural number set and the integer set, they are all  . When expressed in infinity cardinal number, we can distinguish the size of the natural number set and the real number set, because the cardinal number of the natural number set is 0  , and the cardinal number of the real number set is 0 2  = C ; but it is still impossible to distinguish the size of the natural number set and the integer set, they are both 0  .
The second is the calculation of infinity. Whether it is the infinity symbol A or the infinity cardinal number, it cannot play a mathematically precise role in calculations. E.g： The reason for the infinity paradox in mathematical expressions and mathematical calculations is that the existing infinity theory does not need to follow the principle of "the whole is greater than the part", and this principle needs to be followed in the finite number theory. In this way, there is a problem of using different calculation rules in the same calculation formula.
Since there is an infinite problem, how can there be no infinitesimal problem?
For example: because the infinity and the infinitesimals are reciprocal of each other (when the infinitesimal is not zero), the following equation holds: Obviously, in these equations, although mathematical calculations can also be performed, the mathematical accuracy is lost. At the same time, treating zero as a special infinitesimal is inconsistent with the concept of infinitesimals. Because in modern mathematics, the infinitesimal is not a number but a variable, and zero is a specific number, which is inconsistent with the definition of infinitesimal.
It can be seen that the problem of infinity involves many basic mathematics problems, and the mathematics crisis caused by it is no less than the previous three mathematics crises. No wonder Hilbert listed the continuum problem as the top of the 23 mathematical problems.

Grossone method and quantitative calculation of infinity
Sergeyev used Grossone ① to represent the number of elements in set of natural numbers, which is similar to Kantor's cardinal number method. Kantor's cardinal number and Sergeyev's Grossone ① are superficially the same thing. Both represent the size of the set of natural numbers, but they are two completely different concepts.
The cardinal number represents the size of a type of set that satisfies the principle of "1-1 correspondence". For a finite set, the cardinal number is the "number" of elements, but for an infinite set, the cardinal number is not the "number" of elements. Is the size of a class of infinite sets that are equivalent to each other. And Grossone ① represents the "number" of elements in a natural number set, just like any finite set. Using this as a ruler, you can measure every infinity and infinitesimal.
In Grossone theory, infinity and infinitesimal are not variables, but definite quantities. Infinity and infinitesimal are the reciprocal of each other. For example, the number of elements ① of the natural number set is an infinity, and its reciprocal ① 1 is an infinitesimal. Obviously, zero is not an infinitesimal. 4 Let us see how numbers are expressed. The decimal numeral we generally use now are: 1,2,3,4,5,6,7,8,9,0. Among these 10 numeral, the largest numeral is 9, but we can use them to express all finite numbers, whether it is ten thousand digits, billion digits, or larger numbers.
For example, according to the principle of "whole is greater than part", we can get: The Grossone method can not only accurately express infinity, but also accurately express infinitesimal. E.g: For example, infinity can be operated like a finite number: More importantly, the Grossone method solves the calculation problems of   ,  −  , etc. that cannot be performed in the infinity theory.
For example, the following calculations are possible: It can be seen that the Grossone method meets the requirements of the unity of mathematical theory.
From the above discussion, we can see that the cardinal method uses the "1-1 correspondence" principle but violates the "whole is greater than the part" principle, while the Grossone method uses the "whole is greater than the part" principle, but does not violate the "1-1 correspondence" principle. Therefore, the new infinity theory can integrate the infinity cardinal number method with the Grossone method. But when using the infinity cardinal number theory to calculate, we should not use the "=" symbol, but can use "  " to indicate that it is equivalent under the "1-1 correspondence" principle. E.g: However, things are not so simple. Sergeyev also encountered a mathematical problem, which is the "maximal number paradox." Just imagine, if ① represents the number of elements in a set of natural the number of elements in the natural number set is not ①.

Sergeyev thought
, and the number greater than ① is called an extended number [40]. But this is hard to make sense, because 1 ①+ fully conforms to the definition of natural numbers, and the extended natural numbers are still natural numbers. We will discuss this issue later.

Grossone is a number-like symbol used for calculations
Obviously, the number of elements in the odd number set and the even number set is 2 ① , which is less than the number of elements ① in the natural number set.
Sergeyev also created a method of constructing an infinite subset of the natural number set [40]. He Grossone ① is a numeral symbol that represents the number of elements in natural numbers set.
However, the set of integers and real numbers are larger than the set of natural numbers. According to the principle of "the whole is greater than the part", does it mean that there are integers and real numbers greater than ①?
Below we use Grossone method to examine the integer set Z and real number set The integer set and real number set are larger than the natural number set, which refers to the number of elements, rather than the existence of numbers exceeding ① in the integer set and real number set.
In fact, ① is not a number, but infinity. No number can exceed infinity, and ① is a symbol for infinity.
Looking back at the problem of the "maximum number paradox" now, it is not difficult to solve it.
The problem lies in the qualitative aspect of A. In fact, A is just a number-like symbol used for infinity calculations, and is a ruler used to measure all infinity sets. Take 1 ①+ as an example. First, 1 ①+ , like ①, is infinity, not a numeral. Second, It can be seen that the so-called "maximum number paradox" does not exist for Grossone method.

The size of the point is not zero and the continuity of the set
The continuum originally refers to the real numbers set. Since the real number corresponds to the point 1-1 on the straight line, the straight line is intuitively composed of continuous and unbroken points, so the real number set is called the continuum. In the number sequence, the set that satisfies the "1-1 correspondence" relationship with the interval (0, 1) is called the continuum.
Traditional mathematics has an axiom: a point has no size. This can be proved by continuum theory. We use a probability problem exemplified by Sergeyev to illustrate [40]. 8 As shown in the figure above, suppose the radius of the disc in the figure is r , and the disc is rotating. We want to ask a probabilistic event E : What is the probability that point A on the disk stops just in front of the fixed arrow on the right? According to the traditional calculation method, point A has no size, so the probability of occurrence of E is: Although point has a size, point is isolated, so its dimension is zero. We know that straight lines and planes are one-dimensional and two-dimensional continuums, corresponding to real numbers and complex numbers, respectively. There are also three-dimensional and multi-dimensional space continuums, such as Euclidean space, Minkowski space, Riemann space, etc. In this sense, a point can also be regarded as a special continuum, that is, a zero-dimensional continuum.
In Sergeyev's view, the "dense and no-hole" continuum is an absolute continuity. Correspondingly, he proposed a relative continuity theory [40]. In order to distinguish between the two continuums, I call the traditional continuum collectively the absolute continuum, and the set of relative continuity as the relative continuum. is used instead, calculate that the position difference between adjacent elements of the set X is equal to 2 ① , then the set X is discrete in the unit of measure  . Therefore, whether the set X is continuous or discrete depends on the size of the unit of measure  .

Function
) (x f is continuous in the unit of measure at some point ) (x f is said to be continuous in the unit of measure  on set Relative continuum theory provides a new way to solve the problem of continuity of sets other than absolute continuum.

The mathematical continuum is the foundation of the cosmic continuum
The cosmic continuum is a scientific foundation theory based on the mathematical continuum [4]- [14]. The cosmic continuum theory believes that there are three basic entities in the universe: mass body, energy body and dark mass body. Their smallest units are elementary particle min m , elementary quantum min q , and elementary dark particle min d .
In this way, we get three countably-infinite sets in the universe: According to cosmic continuum theory, the coupling energy quantum connects all entities to form the universe as a whole, and any changes in the universe affect the whole. Due to the effect of the coupling energy quantum, the basic existence set E will form the following power set, namely the existence continuum Since space is the existence dimension of the mass body, time is the existence dimension of the energy body, and dark space is the existence dimension of the dark mass body, correspondingly, we The basic dimension set W will correspondingly form the following power set, namely the dimension continuum According to Cosmic continuum hypothesis: the universe is a continuum consisting of an existence continuum and an existing dimension continuum. The existence continuum is composed of mass bodies, energy bodies and dark mass bodies. The existing dimension continuum is composed of space, time and dark space.
Let the cosmic continuum be U , then: In this sense, the cosmic continuum is actually a continuum composed of all physical events in the universe. In other words, every element of the cosmic continuum is a physical event.
Next we examine how to describe physical events in the cosmic continuum.
According to the change axiom of the cosmic continuum: the change of energy is the cause of the state change of all cosmic systems.
Suppose the energy change in a universe system is E  , and the corresponding change in the state of the universe system is In the universe continuum, energy is the independent variable, and the state of the universe system is the dependent variable.
Relative continuum provides us with a mathematical tool for describing physical events.
The change axiom tells us that the essence of all physical events is that energy changes cause the state of the universe to change. Suppose the dependent variable ) (x f is the state of the universe 11 system, the independent variable x is the energy, and S ] , [ b a is the range of the independent variable that can be described by a certain digital system S, then any physical event, including all interactions and quantum phenomena, can be described by the above-mentioned relative continuum . Example 3: Quantum mechanics physical event: photoelectric effect. The dependent variable A is the electronic state, the independent variable B is the photon frequency, and C represents the photon frequency range.

Conclusion
This article discusses several aspects of the development of infinity theory.
(1) Qualitative calculation and quantitative calculation. Cantor used the cardinal number method to solve the problem of comparing infinity. Sergeyev used Grossone method to solve the problem of unifying the calculation rules of infinity and finite numbers.
(2) Absolute continuum and relative continuum. The continuum in traditional mathematics refers to a collection of "dense and no holes", the relative continuum is a continuum that changes with the change of measurement units.
(3) Cosmic continuum and continuum theory. The universe continuum is a mathematical model of existence and its dimensions in the universe. The development of continuum theory provides a new mathematical foundation for the Cosmic Continuum Theory.
The discussion in this article shows that: (1) The theory of infinity will usher in a new era of development. Grossone method is a scientific infinity theory like the cardinal number method; in the new infinity theory, infinity and infinity can be mathematically calculated like finite numbers.
(2) New developments in the theory of infinity will promote the development of fundamental scientific theories. The basic theories of mathematics and physics have always been intertwined and developed, such as classical mechanics and calculus, relativity and non-Euclidean geometry, etc., which are all good stories in the history of science. The relative continuum theory provides a new path for the study of the cosmic continuum.
(3) The essence of Hilbert's first problem "continuum hypothesis" is that the infinity theory is not yet mature. The development of Grossone theory makes this problem self-explanatory. According to the principles of "power set is greater than original set" and "whole is greater than part", there is neither the largest infinity and infinitesimal, nor the smallest infinity and infinitesimal.