Dedeking principlededekind cut , which is the basis for ensuring the continuity of straight lines. Its content is: if all points of a straight line are divided into two categories, so that: 1. Each point belongs to exactly one class, and each class is not empty.2. Each point in the first category is in front of each point in the second category, then, or there is such a point in the first category, and all other points in the first category are in front of it;Or there is a point in the second category, which is in front of all the other points in the second category[3]。This point determines the Dedekin cut of a line. This point is called the Dedekin point (or boundary point). Dedekin's principle was proposed by Dedekin (J.W.) R. Dedekind) in 1872. When constructing the axiom system of Euclidean geometry, it can be selected as the continuity axiom,Archimedean axiomTogether with Cantor's axiom, it is equivalent to DeDekin's principle[1]。
definitionIf theSet of real numbersR is divided into two subsets S and T, which meet the following requirements:
(1)
；
(2)
；
(3)
, there is always x<y (called S as the left set, and T as the right set)
Is called one of the real number set R“dedekind cut ”, recorded as (S, T)[2]。
“dedekind cut ”The first requirement of is that neither the left set S nor the right set T isempty set, which means that there arereal number, which is not empty for short.The second requirement is that S and T contain all real numbers. In other words, for any real number that belongs to either the left set S or the right set T, the two must be in the same place, which is called no omission for short.The third requirement is that the real numbers in the left set S are smaller than the real numbers in the right set T, which is called not messy for short.It can be inferred from Article 3 that the real numbers in the left set will not appear in the right set, and the numbers in the right set will not appear in the left set.If x belongs to the left set, all real numbers less than x also belong to the left set; if y belongs to the right set, all real numbers greater than y also belong to the right set.
For example, order
Readers can verify that (S, T) is a DeDekin segmentation
S={x ∈ R | There is a natural number n, so that
}，
T={x∈R | x≥1}。
This also determines a DeDekin division (S, T).
In the first DeDekin partition, the left set S has the maximum number
, while the right set T has no minimum decimal;The second DeDekin partition is the opposite. The left set S has no maximum number, while the right set T has a minimum number of 1.
And 1 are both called the corresponding intermediate points of DeDeJin segmentation.In general, there must be intermediate points in the Dedekin division of real numbers, which is illustrated by the following theorem, but if a Dedekin division is similarly made on the set of rational numbers, there may not be intermediate points.For example, if S={x ∈ Q | x ≤ 0, or xtwo≤ 2), T={x ∈ Q | x>0, and xtwo>2) Then (S, T) constitutes a pairRational number setThe Dedekin partition of Q, but the left set S has no maximum number;The right set T has no minimum, that is, (S, T) has no intermediate point[2]。
Before constructing real numbers from the set of rational numbers Q by using the Dedekin partition, a lemma is given: between any two rational numbers, there must be countless rational numbers.Lemma is very easy to prove. Let a and b be two rational numbers, then their arithmetic mean value
It must also be rational and c must be between a and b.
Now make a Dedekin partition (S, T) on the rational number set Q at random, and the following three situations may occur.
(1) There is a maximum value in S and no minimum value in T.for example
(2) There is no maximum value in S and minimum value in T.for example
(3) There is no maximum value in S and no minimum value in T.for example
There is no maximum value in (4) S and minimum value in T.This is because if the maximum value in S is a and the minimum value in T is b, according to lemma, their arithmetic mean c is also rational and a<c<b.But because a is the maximum value in S, c is not in S.And b is the minimum value in T, so c is not in T.This leads to the fact that the rational number c does not belong to any set of S and T, which contradicts the requirement of DeDekin's partition that S →T=the complete set Q.
For cases (1) and (2), DeDekin said that the partition determined a rational number, or called such a partition a rational number.For (3), DeDekin said that the partition determined an irrational number, or called such a partition an irrational number.Rational numbers and irrational numbers are collectively called real numbers and recorded as R. Therefore, each real number is a partition of the set of rational numbers Q.
Under this definition, we can give the definition of real number equality and compare the sizes.
Equality: Let the real numbers a and b be two Dedekin partitions (S, T) and (S', T '). If the set S=S' (T=T' must exist at this time), then it is called a=b.
Size comparison: if the set S ⫋ S', it is called a<b.If the set S ⊆ S', then a ≤ b.
That is to say, to prove that two real numbers are equal, it is only necessary to prove that S and S' obtained from the division are equal.
Dedekin theorem
Announce
edit
Any Dedekin partition (S, T) of the set R of real numbers uniquely determines a real number
(called intermediate number or intermediate point), which is either the maximum number of S (there is no minimum number in T) or the minimum number of T (there is no maximum number in S).
