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Definition of non number: if the element δ and any element ε cannot satisfy δ ε, then δ is non number
Author: Xie Ling
ORCID ID: 0000-0002-1957-603X
Address: College of cosmic mathematics and physics, China / Hunan
E-mail: 29997609@qq.com Mobile phone: 13973920402
Abstract:
Using two interesting experiments, the infinite element is regarded as a number, the mathematical logic operation of number is carried out, and the wrong conclusion is obtained.
I use logical relations to define number and non number, finite and infinite.
Based on the definition, it is proved that all infinite elements are not numbers.
The reasons for the errors in two interesting experiments: the mathematical logic operation of non numbers.
Found the root causes of the three major crises in mathematics.
Solution: non numbers cannot be used for mathematical and logical operations.
Key words: Limited concept; The concept of infinity;number; non number
Mathematical classification code: 03B05 ; 03B22;03B38;11U09;11-03
research method
Use scientific standards:
① Every concept must be defined;
② Every definition must be logical (please prove that your definition is logical, or use logic to define it);
③ Each definition has its own symbol representation (reasoning and argumentation only recognize symbols);
④ It can't conflict with the correct definition of predecessors and ancients;(those conforming to the first three items are the correct definitions).
⑤ All the relations of definitions constitute a manuscript, which must be composed by logic;
⑥ All argumentation and refutation can only refer to the concept with logical definition, and can not introduce new concept (new concept: the concept without logical definition).
According to the six standards of science, all logical theories will not conflict and paradox, and mathematical (all scientific) logic is self consistent.
To do the above six is scientific behavior, not to do is pseudo science (Sorcery)
In order to prevent pseudoscience from quarreling, we must abide by the sixth principle.
Attachment: logical (a ≯ a), illogical (a ＞ a).
Got: scientific standards are fair truth (because: logical).
Evil ideas against scientific standards always exist, because scientific standards challenge evil ideas.
Opponents of scientific standards will argue that every concept goes back to "the most primitive concept (commonly known as the atomic concept). How do you ensure that the original concept definition is logical?".
answer:Because it is the most primitive concept and there are no other concepts, the most primitive concept will not conflict with other concepts (see logical definition. Logical equivalent expression: no contradiction; No conflict.).
So: the original concept is instinctive and natural, logical.
The refuter's Refutation is null and void.
research objective
The definition and proof of digital concept is the foundation of mathematical construction.
Because this manuscript defines the concept of numbers,
Let human know what is number, strengthen the theoretical foundation of mathematical edifice.
It eliminates all the conflicts in mathematical theory.
Based on the construction of mathematical theory, the most primitive concept is logically defined.
Eliminate all mathematical contradictions and mathematical paradoxes.
Standardize the logical definition of mathematical concepts.[1][2][5]
Because only I can define finiteness and infinity by logical self consistency.
Through the definitions of finite concept, infinite concept and number concept, it is proved that infinite concept is not a number.
introduce
Interesting case[3]:
Take an infinite element: 0.999......
Infinite element: 0.999......
Assuming that the infinite element is a number,
That is, suppose 0.999...... Is a number,
If: 0.999...... = 1.( Note: unlimited 9).
Or: lim 0.999...= 1.( Note: unlimited 9).
Logic: only numbers can be associated with mathematical symbols. Non numbers cannot enter the mathematical system.
(Ⅰ)1=0.999...
∵ 1=1
∴ The first time, model change: recursive descent mode. Recorded as:1=0.999......
∵1=0.999...
∴1=(0.9)+(0.09)+(0.009)+...
The second time, model change: recursive descent mode.
∴1=(0.89...)+(0.089...)+(0.0089...)+...
∴1=(0.8+0.09+0.009+...)+(0.08+0.009+0.0009+...)+(0.008+0.0009+0.00009+...)+...
The third time, model change: recursive descent mode.
1=(0.79...+0.089...+0.0089...+...)+(0.079...+0.0089...+0.00089...+...)+(0.0079...+0.00089...+0.000089...)+...
Infinite times (n→∞), model change: recursive descent mode.
1=(0.000...+0.000...+0.000...+...)+(0.000...+0.000...+0.000...+...)+(0.00...+0.000...+0.000...)+...
→ 1=0.
Is a wrong conclusion.
(Ⅱ) 1=lim 0.999...
The same logic leads to:1=lim 0.
Is a wrong conclusion.
→ 1=0.
Is a wrong conclusion.
(Ⅰ)+(Ⅱ) →The concept of infinite convergence is regarded as a number and a wrong conclusion is obtained.
Interesting case[4]: 1+2+3+4+5+... = -1/12
prove:
Infinite natural sequence:1,2,3,4,5,...
Infinite natural sequence in the form of addition and summation(A):1+2+3+4+5+...
If infinite (A) is a number,
Mathematical logic can make: 1+2+3+4+5+... = S
Take: S1=1-1+1-1+1-1+...
→ 1-S1=1-(1-1+1-1+1-1+...)=1-1+1-1+1-1+...=S1
→1=2S1
→S1=1/2
Take: S2=1-2+3-4+5-6+...
→2S2=1-2+3-4+5-6+...
0+1-2+3-4+5-6+...
=1-1+1-1+1-1+...=S1
→2S2=S1 →S2=S1/2 →S2=1/4
S-S2= 1+2+3+4+5+6...
-(1-2+3-4+5-6+...)
=0+4+0+8+0+12+0+...
=4+8+12+16+20+...
=4(1+2+3+4+5+... )
=4S
→-S2=3S →S=-1/12. Is a wrong conclusion.
→Take the concept of divergent infinity as a number and get the wrong conclusion.
Two interesting cases →The concept of infinity as a number leads to the wrong conclusion.
What logic takes "the concept of infinity as a number" and comes to a wrong conclusion.
Why?
Must be: definition of number and non number; Definition of finite and infinite.
Two interesting cases : Finite concept = infinite concept. (0.999...=1)
If we prove that the "infinite concept" is a non number with a logical definition, we get: number ≠ non number.
Interesting example1: the "contradiction" in Cantor's "diagonal method [4]" directly proves that the infinite element is not a number.Cantor did not prove that real numbers were uncountable.
Because Cantor used two assumptions: Hypothesis 1: infinite elements are numbers; Hypothesis 2: real numbers are countable.
He found contradictions in the mathematical derivation of the "diagonal method", which can not be attributed to "hypothesis 2".
∴0.999...≠1 ,∴0.333...≠1/3
I will explain the source of errors in these two interesting cases with mathematical basis and logical definition (axiomatization).
The first concept of mathematics is the concept of numbers
Once the concept of number is defined, number gives logical meaning to number operation.
At the same time, mathematical theory was born.
We can't define numbers with the concept of numbers,
You can't use mathematical theory to define numbers,
You can't define numbers with sub concepts of numbers,
It's like you can't use the concept of plow to define the concept of cow (Logic: there's a cow before there's a plow).
You cannot define numbers with sets because "non numbers" can also be sets.
The digital system cannot be used to define numbers,
{∵ define number} → {(various numbers) + (multiple number system)}.
Existence theorem of non number concept: the universe cannot be all numerical concepts. If there is a numerical concept, there must be a non numerical concept.
prove:
The concept of taking a number is: N
Get the concept of non number: N
Get: NN
Hypothesis: there is no non digital concept in the universe. →{NN}. It contradicts the front.
QED.
(Tip: the definition of digital concept must be limited by non digital concept. If not, it is a circular definition.So I found out how to define logically: finite and infinite must be defined at the same time; Real and empty must be defined at the same time; Numbers and non numbers must be defined at the same time; Rational and irrational numbers must be defined at the same time.
In this way, the pseudo scientific concept can be revealed.)
Definition
Is number a finite concept? Number or finite concept? Number or (finite, infinite) concept?
Therefore, it is necessary to define the concept of finite and infinite. Human beings first have the concept of finite and then the concept of infinite.
finite concept definition: a ...p
In today's language:start with the first element a that you identify select, and then arrange the elements you mark in order until you want to and you can stop at an element p.
Explanation: if we only discuss a to p, then p is the last element from a to p.
[Note: you go from a to p. no matter how many elements you pass through, even astronomical numbers, when you get to p, you get a finite number of elements between a and p. You go through the middle in turn to get to p. all the elements you can reach are finite. It is found that there are finite elements between a and p, that is, a to p is finite. When you take the last p,The second element q in the final order becomes new (a ...q), because you go through q to p in turn. ]。
Therefore, any finite element k and {a ...p} Can satisfy: a ...pk
Therefore, any finite element “=” and {a... P} Can satisfy: a ...p=
Note: three points are used in the finite {a... P} symbol, and the expression is finite, writable and ergodic. Three dot ellipsis {...} It is used because it is long and many.
infinite concept (element) definition:a ......
In today's language: starting from the first element a , you set the program: the elements are arranged in a single column in sequence, and can not be ended. There is no element you are willing to terminate (there is no last element).
Summary: from the first element a , the elements are arranged in order, and there is no last element.
[Note: the ellipsis of the finite symbol a ...p is a finite number. If you remove p, there will be the last one from the bottom of the new sequence.
a ...p The middle ellipsis belongs to the limited complete writing, which is omitted because of its lengthy.
Infinite symbol a ......
(1) has no last element and belongs to an endless array of elements. In order to distinguish from the finite, the symbol uses 6 points:......
(2) cannot be supplemented with any symbol k at the end. The infinite symbol is not allowed to be: a ......k.
Because the concept of infinity does not allow the last element, there is no "position" of the last element. ]
It is concluded that the above definitions of finite and infinite generalizations are defined by logic.
When we define finite and infinite by logic, we can't refute it with real infinity and latent infinity.
Because there are finiteness and infinity first, then we can define real infinity and latent infinity with finiteness and infinity.
In the definition of finite and infinite, the concept of real finite and potential infinite has not yet been born.
How to define numbers?
When numbers were not defined, all mathematical theories were not born.
Therefore, we can't use mathematical theory to define number, let alone refute my definition of number with mathematical theory.
The concept of number belongs to the original concept and will not conflict with the correct definition of predecessors.
(original concept: the first concept to create a group.
Therefore, the original concept can be proved to be logical, logical definition: it does not conflict with other concepts and is self consistent.
The concept added later is not allowed to conflict with the correct definition, so the original concept definition is always logical.)
In this way, numbers can only be defined by logic.
Definition of the concepts of number and non number
Definition of non number: if the element δ and any element ε cannot satisfy δ ε, then δ is non number
Logical self consistency gives the definition of number: if the element α and any element β satisfy α β, then α is a number.
According to: as long as the element δ and any element ε cannot satisfy δ ε.
We get that: δ cannot satisfy: δ =,
We get that: δ cannot satisfy: δ ＜,
We get that: δ cannot satisfy: δ >,
It is proved that δ can only exist alone.
Logically, δ is a non number.
Logical proof
theorem: Infinite elements (Concepts) are not numbers.
prove:
Take any finite element K: ∀K,
According to the definition of finite element, the finite element "K" can be extended to K β.
From the definition of number:
K can satisfy: k =, K ＜, K ＞.
Take any infinite element (1): ∀a......
It is defined by the concept of infinity (infinite element): the concept of Infinity has no last element,
It is known from (1) that any finite element K is not allowed to satisfy: ∀a... ...K,
According to (1), it is not allowed: ∀a......＝, not allowed: ∀a...... ＞, not allowed: ∀a......＜,
Defined by infinite elements, infinite elements: a......
Not allowed: a... β.The result shows that the infinite element can only exist alone and can not be extended with any element.
It is proved that any infinite element is non number.
The same logic is derived from (1): not allowed to:∀a......k{＝,＜，＞}。
We obtain that (1) is a non number: Infinite elements (Concepts) are not numbers.
QED.
 
My paper above proves to be complete.
My goal is to achieve, and my argument is in line with mathematical logic.
conclusion
It is proved that number can only be constructed in finite steps, and infinite number is a concept that cannot be completed. Because Infinity has no last element.
π is a number, 3.14159 is a number, 3.14159 ...7 is a number,
This is not a number: 3.14159......
π≠3.14159......
is number, 1.414 is number, 1.414... 3 is the number,
This is not a number: 1.414......
≠1.414 ......
It is proved that: 1+2+3+4+5+... = -1/12, it's wrong.
It is proved that 1/3=(0.1)3≠0.333.....
(Tip: 1 / 3 definition (see Euclidean geometry drawing) shows that it is a "ternary number system", which cannot be processed with "decimal". Therefore, using "decimal" to deal with it is a violation of the definition.)
Take the concept of Infinity: 1+(1/2)s+(1/3)s+(1/4)s+(1/5)s+......
Logical definition cannot meet: 1+(1/2)s+(1/3)s+(1/4)s+(1/5)s+......=ζ(s)
Logical definition allows:1+(1/2)s+(1/3)s+(1/4)s+(1/5)s+...+(1/n)s=ζ(s)
So: the inverse mathematical logic of Riemann ζ function.
That is to say, the Riemann ζ function with infinite prime numbers is a pseudo concept.
Because the "infinite concept" is not a number, the first two interesting cases are explained.
It proves that modern mathematics has made mistakes in the basic theory from the source. You can't refute me with mistakes. All your refutations are wrong.
Remember: axioms, definitions must be logical. Please prove that the concept definition you brought is logical.
Intuition: (finite ≠ infinite), We can't rely on intuition to say it's right. We must prove it logically.
Intuition: (0 = 0 + 0 +0+...... Or 5 = 5.000......), You can't intuitively say they're right. Logic proves them wrong: real numbers are defined as the distance between two points (lines and curves between two points. Note:between. between two points, points are not included.) Zero dimensional points (0) do not form distances (No arbitrary lines and curves are formed).
The limit of Cauchy function (ε-δ definition)[8] is a self contradictory definition, which I exposed in another paper.
Statement:
Funding (supported by the author)
Conflict of interest / competitive interest (none)
Availability of data and materials (none)
Code availability (none)
Ethical approval (APPROVAL)
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Reference
[1]The Commutative Law of Addition, and Infinity
PHILIP E. B. JOURDAIN;Research 15 Jul 1909；《Nature》Volume: 81, P: 69
[2]The Philosophy of Bertrand Russell
EDMUND T. WHITTAKER;Books & Arts 03 Feb 1945；《Nature》Volume: 155, P: 128-131
[3]Interesting case
http://www.mathchina.com/bbs/for ... 1&extra=page%3D
[4]Interesting case:Wikipedia: Cesaro seeks the sum of natural numbers
http://zh.wikipedia.org/wiki/%E5 ... 7%E6%B1%82%E5%92%8C
[5]Cantor was able to show that the real numbers can’t be put into a one-to-one correspondence with the natural numbers(Cantor diagonal rule)
KEVIN HARTNETT；SEPTEMBER 12, 2017；
https://www.quantamagazine.org/m ... yre-equal-20170912/
[6]Cauchy mathematical limit:The Basis of Modern Mathematical Analysis
L. M. M-T.Books & Arts 31 Aug 1935；《Nature》Volume: 136, P: 315-316
[7]Cauchy's convergence test:Theory of Functions
H. T. H. PIAGGIO；Books & Arts 11 Aug 1951；《Nature》Volume: 168, P: 216-217
[8]Limit of a function (ε-δ definition)
https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/ |
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