Is to meet the following conditions
A. A *** N
Elemental nitrogen
And the successor element n+
N+ corresponds to N.B. Element E must belong to N.C. Element E is not the successor of any element in N.D.N.
A+=b+ Then A = B. (unique element) E. (inductive postulate) S is a subset of N.
E belongs to S.
N belongs to s
N+ also belongs to S. Then S=N. N is what we call a natural number * * *. We specify e:= 1.
e+:=2
(e+)+:=3
..... and so on. 2. Then define addition.
Addition (+) is a function.
This function satisfies two conditions: 1. (+)(n
E)=n+ Write the familiar formula 1.n(+)e=n+ 2. (+)(n
m+)=((+)(n
M ))+ 2.n(+)m+=(n(+)m)+ Function (+) satisfying the above conditions.
We call it addition+. (+): =+A function that satisfies these two conditions can be proved to exist and be unique: The proof is as follows because (+)(e
E) = e+e (+) e = e+so1+1= 2 is proved. Existence: e.
e+
(e+)+
..... that all natural numbers are unique: n N "? ，+(n
e)=n+ +(n
e+)=(+(n
e))+ +(n
E+)+) = …………………………。 And these 1+, (1+)+, ... are represented by symbols 2, 3, ... so 1+ 1 refers to the number after 1, that is, 1+. Why is there Peano postulate and definition addition? This originated from hibbert and Brouwer at the end of19th century and the beginning of 20th century. Because of the special relativity and quantum theory in physics, mathematicians want to prove that mathematics has a solid foundation and is an unchangeable truth. Therefore, I hope to establish a complete and rigorous foundation in logic, so I started with the natural number system, hoping to get the complete nature of the natural number system from the basic postulates like Euclid geometry, so I summarized Peano's five postulates (in fact, later generations further simplified them into three), and Russell and his teacher Whitehead co-wrote < < Mathematical Principles > > > Three volumes are only part of the work. Hilbert drew up a series of plans to transform the basis of mathematics into logic, so that mathematicians can declare that "mathematics is truth". Unfortunately, in 1929, when Godel was 23 years old, he proved a theorem: the incompleteness theorem: if a system contains arithmetic and the basic assumptions of this system are not contradictory, then there must be a proposition in this system, and neither the affirmation nor the negation of this proposition can be proved. So mathematics is not just logic. Of course, we can see whether the proof of "1+ 1 = 2" is meaningful from Godel's theorem. Simple method: 1+ 1=2. . . ( 1+ 1)- 1=2- 1。 . . 1= 1 Save1+1> 2。 . . ( 1+ 1)- 1 & gt； 2- 1。 . . 1 & gt； 1 is not valid. 1+ 1 < 2。 . . ( 1+ 1)- 1 & lt； 2- 1。 . . 1 & lt； 1 is invalid.
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1+ 1=2 Don't underestimate this formula. 1+ 1=2 is one of the greatest formulas in science. Many people may ask, "Why 1+ 1=2?" This seems to be redundant (! ? ) problem. Now I'm trying to briefly introduce to interested netizens how to prove the mathematical statement "1+ 1=2" within the framework of axiomatic theory, which is "irrefutable" for most people. First of all, we should know that in the context of * * * theory, the objects we are discussing are all kinds of * * * (or classes, and the difference between them and * * * is not superfluous here), so the natural numbers we often encounter are also defined by * * * (or classes) here. For example, we can define 0, 1 and 2 (for example, qv. Quine) in the following way.
mathematical logic
Revised edition.
Chestnut 6
43-44):0:= { x:x = { y:~(y = y)} } 1:= { x:y(yεx . & amp； . x \ { y }ε0)} 2:= { x:y(yεx . & amp； . X \ {y} ε 1)} [For example, if we take an element from a molecule that belongs to the category of 1, then this molecule will become zero. In other words, 1 is a class composed of all classes with only one element. Now we generally use the method mainly introduced by von Neumann to define natural numbers. For example: 0: = λ
1:= {Λ} = {0} =0∪{0}
2:= {Λ
{Λ}} = {0
1} =1∨ {1} [λ is an empty set] Generally speaking, if we have constructed the set n,
Then its successor n * is defined as n ∨{ n}. In the general axiom system of * * * (such as ZFC), there is an axiom to ensure that this construction process can continue continuously, and all * * * obtained by this construction method can form a * * *, which is the so-called infinite axiom (of course, we assume that other axioms (such as the union axiom) have been established. Note: Infinite axioms are some so-called illogical axioms. It is these axioms that make some propositions of the logician school represented by Russell impossible in the strictest sense. ] use? We can apply the following theorem to define the addition of natural numbers. Theorem: Life "|N" means that * * * consists of all natural numbers, so we can uniquely define the mapping A: |NX |N→| N, so that it satisfies the following conditions: (1) For any element X in | n, we have a (x.
0)= x； (2) For any element X and Y in |N, we have a (X.
y*) = A(x
Y)*. Map A is the map we use to define addition. We can rewrite the above conditions as: (1) x+0 = x; (2) x+y* = (x+y)*. Now we can prove that "1+ 1 = 2" is as follows: 1+ 1 = 1 * (because1:. ] 1+ 1= 2 "can be said to be a" natural "conclusion drawn by human beings after introducing natural numbers and related operations. However, it was not until the19th century that mathematicians began to establish a strict logical foundation for the analysis based on real number system, and people really examined the basic problems about natural numbers. I believe that the most "classic" proof in this respect should be the one that appeared in "Principles of Mathematics" co-authored by Russell and Whitehead. We can prove "1+ 1 = 2": First, we can infer: α ε 1
y}。 & amp。 ~(x = y))ξε 1+ 1 & lt； = & gt(σx)(σy)(β= { x } ∨{ y }。 & amp。 ~(x=y)) So for any * * * γ, we have γ ε 1+ 1
y}。 & amp。 ~(x = y))& lt； = & gtγε2 According to zermelo-fraenkel of * * * theory, we get 1+ 1 = 2. ]
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