Piano's five axioms are described informally as follows: ① 1 is a natural number; ② Every certain natural number A has a certain successor number A', A'
It is also a natural number (the successor of a number is the number immediately after this number, for example, the successor of 1 is 2, the successor of 2 is 3, and so on. ); (3) If both B and C are successors of natural number A, then B = C；; ;
④ 1 is not the successor of any natural number; ⑤ Any proposition about natural numbers can be proved to be right for n' if it is proved to be right for natural number 1 and it is assumed to be true for natural number n..
Then, this proposition holds true for all natural numbers. (This axiom is also called inductive postulate, which ensures the correctness of mathematical induction. If 0 is regarded as a natural number, then 1 in the axiom should be replaced by 0. A more formal definition is as follows:
A Dai Deking-piano structure is a triple (x, x, f) that satisfies the following conditions: X is a set, X is an element in X, F is a mapping of X to itself, and X is not in the range of F. 。
F is injective. If x∈A is satisfied, if a∈A is satisfied, then f(a)∈A is a = X. This axiom and the basic assumptions about the set of natural numbers derived from Piaro's axiom:
1.n (natural number set) is not an empty set.
2. There is a one-to-one correspondence between the direct successor elements from A to A in n..
3. The subsequent element mapping image set is a proper subset of n..
4. If any subset of p contains both non-successor elements and successor elements, and the successor elements contain every element in the subset, then this subset coincides with n. 。
Can be used to demonstrate many theorems that are common at ordinary times but don't know the source! For example, the fourth hypothesis is the theoretical basis of the widely used first inductive principle (mathematical induction).
It is proved that the successor number of 1+ 1 is the successor number of 1, that is, according to Piano's axiom 32, the successor number is 3.
④
Available: 1+ 1=2
Contrary to even numbers and even numbers, the unity of opposites, arithmetic axiom 1+ 1=2 and why1+kloc-0/= 2 are comprehensive contradictions that belong to both philosophical and mathematical categories, and dialectics of nature (philosophy) and mathematics are inevitable, …, why1+6544. Odd and even numbers are unity of opposites, and subsets are derived in the system, that is, in the process of development and change (taking positive numbers as an example only), the scores are 1/2, 3/2, 5/2, 7/2, 9/2, 1658. 3.5, 4.5, 5.5, 6.5, ... have been differentiated to occupy the position of integer, which fully and completely reflects the philosophical integrity of the score, and the score is odd 1, 3,5,7,9,1,13,6544.
..... is an axiom, and 2 is the first axiom of the mathematical axiom system. Obviously, integers form generalized integers, number theory forms generalized number theory, set theory forms generalized set theory, and truth forms generalized mathematical truth, which lays a solid foundation for quantum mechanics and reveals some laws of motion (spin) of basic particles such as protons, neutrons and extranuclear electrons in atoms of the universe, namely fermions and bosons, generalized integers and bosons. …。
Keywords: 1, odd number, 2, even number, 3, opposition, identity, 4, philosophical integer fraction or philosophical integer fraction, 5, philosophical integer property, 6, law of unity of opposites, 7, derivative subset, 8, why 1+ 1=2, 9, generalized.
1, even and odd numbers contain odd laws in the philosophical and mathematical sense: if the mathematical contradiction between even and even numbers is discussed from the perspective of natural dialectics (philosophy) and mathematics, the traditional mathematical theory that even numbers are divisible by 2 and odd numbers are not divisible by 2 only involves the opposition, exclusion and difference between even and odd numbers, but does not involve the similarities and differences (contradictions) between even and odd numbers and the commonness and identity in the differences. Obviously, it is an incomplete rational understanding with one-sidedness ... If odd and even numbers are a pair of philosophical contradictions with mathematical significance, then the two aspects of this contradiction are not only different and different, but also identical-the same in the difference and the same in the difference. If there are commonalities and identities in differences, we must explore and seek scientific basis, and we must not make them out of nothing. Dialectics of nature (modern philosophy) and dialectical mathematical logic jointly find that in the axiomatic system of mathematical logic, the scores of derivative subsets (only for positive examples) are 1/2, 3/2, 5/2, 7/2, 9/2,1/2,650. Therefore, it is different from the development of numerical logic system and occupies the position of integer, which fully embodies the philosophical integrity of its fraction. Why does it have the philosophical integrity of fraction? Because 1/2 is the largest decimal unit, in other words, the decimal (just taking positive examples) is 0.5, 1.5, 2.5, 3.5 and 4.5. The absolute value of ... is relatively complete compared with the absolute values of other decimals (don't be confused by the phenomena and illusions of its decimal nature), so it is different from the development and changes of the system, occupying the position of an integer, that is, the derivative subset acts as an "integer", which fully and completely reflects the philosophical integrity of its decimal, and the system has complete axioms 2, 3, 4, 5, 6, 7 and 8. There are multiples of 1 1, 12, 13, 14, 15, 16, ... or even links of the system have 2, 4, 6, 8. Axiom 1 1, 13, 15, 17, ... 2 is the first axiom of a mathematical system, and its philosophical integrity is that odd numbers (including prime numbers) can be divisible by 2, which provides science for the contradiction between odd numbers and even numbers in philosophy and mathematics. Therefore, dialectics of nature (modern philosophy) opens the way and points out the correct direction for how to correctly answer the mathematical truth of 1+ 1=2. So even numbers can be divisible by 2, odd numbers can't be divisible by 2, and odd numbers (including prime numbers) can indeed be divisible by 2. They are not only antagonistic, exclusive and different, but also common and identical. In other words, odd and even numbers are the same, and there is a dialectical relationship of unity of opposites. Odd number and even number are both a pair of contradictions with philosophical connotation and a pair of contradictions with mathematical significance. Therefore, we need dialectical analysis and dialectical reasoning, the guidance of dialectics of nature, and the full support of mathematical experts and philosophical experts. We should break through the shackles of traditional mathematical thinking concepts and traditional classical number theory and set theory.
2. Philosophical integral score, philosophical integral decimal, philosophical integral nature: the score is 1/2,-1/2, 3/2, -3/2, 5/2, -5/2, 7/2, -9/2, -9/2. The decimals are 0.5, -0.5, 1.5,-1.5, 2.5, -2.5, 3.5, -3.5, 4.5, 5.5, -5.5, ... and their philosophical integer properties are collectively called philosophical integer decimals and philosophical integers. What is philosophical integrity? That is, the absolute values of other decimals (other fractions) are more dispersed than those of philosophical integer decimals (philosophical integer fractions). In other words, the absolute value of philosophical integer decimal (philosophical integer fraction) is relatively complete than that of other decimals (other fractions). This relative integral property is similar to the integral property of integers in differences, and the commonness and identity in this difference are collectively called the philosophical integral property of decimals (the philosophical integral property of fractions). Although it is relative, it also exists objectively. The philosophical integrity of decimals provides an objective scientific basis for odd numbers to be divisible by two philosophies, which is a great discovery and victory of dialectics of nature! This is a question of understanding the world outlook. Obviously, philosophical integer decimal (philosophical integer fraction) has contradictory dual nature: one is philosophical integer nature, and the other is ordinary decimal (ordinary fraction). Only philosophical integer decimals (philosophical integer fractions) have philosophical integer properties, and other ordinary decimals (other ordinary fractions) do not have philosophical integer properties, because 1/2 is the largest fractional unit.
3. Philosophical and mathematical significance of philosophical integer decimals: Philosophical integer decimals provide identities for odd and even numbers, and odd numbers can be divided by 2.
The philosophical divisibility provides a scientific basis for the mathematical truth 1+ 1=2. Odd number and even number are a pair of comprehensive contradictions, which belong to both philosophical and mathematical categories. Integer and philosophical integer decimals can be divisible by 2 even numbers, which provides a complete scientific basis for odd numbers to be divisible by 2 philosophy. It seems impossible to understand and accept them correctly from a simple mathematical point of view. Idioms complement each other. Mr. Lao Tzu proposed it more than two thousand years ago. Everything on the contrary has identity, and the contradiction between odd and even numbers is no exception. The philosophical and mathematical significance of philosophical integer decimal is mainly to provide scientific basis for odd numbers to be divisible by two philosophies, and to provide common identities for odd numbers and even numbers. Philosophy (dialectics of nature) points out the correct direction for complete mathematical truth!
4. Scientific basis and source of philosophical integer (fractional) decimal: Obviously, philosophical integer (fractional) decimal has contradictory duality: one is philosophical integer, and the other is ordinary (fractional) decimal. Fractions have decimal units, 1/2 is the largest decimal unit, decimals have decimal units, and 0.5 is the largest decimal unit. The maximum number unit "0.5" and its derivative subset in dialectical mathematical logic provide scientific basis for the philosophical integer decimal (philosophical integer fraction) to have philosophical integer properties. So even numbers are divisible by 2, and odd numbers are not divisible by 2. If it is extreme and absolute, the identity between odd number and even number will be ruled out, that is, if the odd number can be divided by 2 in philosophy, it will hinder the development and breakthrough of complete mathematical truth. As the foundation of mathematics for thousands of years, the development history of mathematical logic itself has fully proved this point. The traditional theory that even numbers are divisible by 2 and odd numbers are not divisible by 2 does not answer the mathematical truth why 1+ 1=2, and the axiomatic system of mathematical logic has not been established. This is because odd numbers cannot be divisible by 2, and it is theoretically impossible to directly admit and accept that 2 is a mathematical axiom. This is also a great pity for (mathematical) arithmetic, because the traditional systems of classical number theory and set theory only have odd rings and no even rings, in other words, only odd axioms and axioms without even multiples 2, 4, 6, 8, 10, 12, 14,/kloc-0. Because they can't completely replace the great significance and function of mathematical logic and its budget law, the traditional theory that even numbers can be divisible by 2 and odd numbers can't be divisible by 2 only knows half of the complete mathematical truth, and only involves contradictions and differences. It was formed in the Pythagorean period, and the other half, that is, the identity of contradictions and the commonality of similarities and differences-the philosophy that odd numbers can be divisible by 2, is also necessary and important. Obviously, number theory and set theory broke through the traditional classical number theory and set theory, and formed generalized integer, generalized number theory and generalized set theory, and truth formed generalized mathematical truth. Generalized integers have laid a solid foundation for quantum mechanics.
5. Odd and even numbers contain the law of unity of opposites in philosophy and why the mathematical truth is 1+ 1=2. In this paper, the mathematical contradiction between odd number and even number with philosophical connotation is simply summarized as follows: even number can be divisible by 2, odd number can be divisible by 2, and odd number (including prime number) can indeed be divisible by 2. Philosophically, odd number and even number are not only antagonistic, but also exclusive and different. That is, there are similarities in differences, similarities in differences, even numbers divisible by 2 and odd numbers divisible by 2 are philosophical similarities in differences and similarities in differences. Even numbers divisible by 2 and odd numbers divisible by 2 refer to the opposition, exclusion and difference between even numbers and odd numbers. So there is a dialectical relationship between odd number and even number (integer and philosophical decimal), which reveals that 2 is the first priority of mathematical axiom system. Dialectics of nature and mathematics are integrated and dialectical unity. This is a question of understanding the world outlook. What kind of world outlook has what kind of epistemology and methodology? Why 1+ 1=2? Our answer is simple and profound: even numbers are divisible by 2, odd numbers are divisible by 2, and odd numbers (including prime numbers) are indeed divisible by 2. Philosophically, odd numbers and even numbers complement each other. Why 1+ 1=2? It is really simple and profound. On the surface, it seems to be the basic knowledge of primary school students, but in fact it is profound and even unreasonable, and it is difficult to understand and accept. There are too many reasons in the world. Why is there no mathematical truth so far? Why publish 1+ 1=2? Is it that it does not exist objectively or that we earthlings have not formed a rational understanding of it? This paper gives an exploratory answer to this question. Please forgive me for the inappropriate points.
6. The axiomatic system of dialectical mathematical logic (generally expressed in the following form, without further expansion, the symbol: ↓ means to assign a group of children): {[0 ~1]}1↓ {[1~ 2]} 3 ↓ {[2 ~ 3]} 5.
(This structural formula is staggered up and down, Corresponding to non-scattering) {[0.5 ~1.5]} 2 {[1.5]} 4 {[2.5 ~ 3.5]} 6 ... link 1: 1 ∑ {second link: 2 ∑. Sixth link: 6 ∑ {[0 ~ 1]} = ∑ {[2.5 ~ 3.5]}, seventh link: 7 ∑ {[0 ~ 1]} = ∑ {[3 ~ 4]}, and eighth link: 8 ∑ {[
7. Generalized integers and generalized mathematical truths have laid a solid foundation for quantum mechanics: integers and philosophical integer fractions are collectively called generalized integers, namely 0, 1/2,-1/2,-1, -65438+3/2, -2, -2. 11/2,-11/2,6,13/2,-13/2,7,/kloc-0.
,-0.5,1,-1,1.5,-1.5, 2,-2, 2.5,-2.5, 3,-3.5,-4,-4. It shows the spin laws of atoms, neutrons, protons, extra-nuclear electrons and other particles, fermions and bosons in the micro-world of the universe. The numerical logic law of the unity of opposites between integers and fractional semi-integers (fractional semi-integers) reveals that both the macro-world and the micro-world contain the law of unity of opposites, which is the universal law of the universe, and the spin motion law of fermions and bosons also contains the law of unity of opposites. For example, the spin law of fermions follows 1/2, 3/2, 5/2, 7/2, 9/2, 1/2 respectively, and the bosons follow 0,12 respectively. Semi-integers in quantum mechanics provide objective scientific evidence and objective support for generalized integers and generalized mathematical truths ... The potential is infinite, and generalized integers and generalized mathematical truths do come in handy. Generalized integers reveal the spin laws of elementary particles, such as protons, neutrons and extranuclear electrons, in the micro-world of the universe, that is, fermions and bosons, and the law of unity of opposites in the numerical logic of integers and philosophical integral fractions (philosophical integral fractions) reveals that both the macro-world and the micro-world in the universe contain the law of unity of opposites, which is the universal law of the universe. For example, the law of spin motion of fermions and bosons also includes the law of unity of opposites ... In quantum mechanics, such as (n+ 1/2) or (Z+ 1/2) is called a semi-integer, and quantum mechanics is a generalized integer. Generalized integer and generalized mathematical truth have laid a solid foundation for quantum mechanics. Obviously, semi-integer has not formed a complete rational understanding in quantum mechanics. The so-called "semi-integer" is actually the so-called "philosophical integer fraction or philosophical integer fraction", which belongs to the category of generalized integer and generalized mathematical truth. So far, generalized integer and generalized mathematical truth have possessed objective scientific evidence, and the mathematical truth in this paper is not the so-called empty talk about mathematical theory. Mathematical truth has many application values, and why 1+ 1=2 is no exception, and why 1+ 1=2 is both a mathematical truth and a major contradiction in mathematics. Solving the main contradiction in mathematics is the primary task and mission of mathematics, which should be solved at the beginning of mathematics. Unfortunately, there was no dialectics of nature and dialectics at that time. At that time, the philosophy adopted by people was not good, which led to the contradictions and problems in mathematics that have been preserved to this day. Why 1+ 1=2 belongs to the category of arithmetic problems and arithmetic, and belongs to "pediatrics" in front of experts. Although it belongs to "pediatrics", as the saying goes, the simplest, simplest and most basic is precisely the most profound, and mathematics (arithmetic) has been fulfilled. The simplest mathematical logic contains the deepest mathematical truth, the law of unity of opposites, the first axiom of mathematics, and the philosophical integrity of fractions (the philosophical integrity of decimals) is the "winding" of arithmetic (mathematics), which is the most difficult mathematical knowledge and truth to understand and accept. Mathematical logic and formal logic cannot reason and prove the philosophical integrity of the fraction 1/2 (decimal 0.5).
Well, please adopt it.