“Dedekind cut”的概念、定义、翻译、参考文献-科学参考

Dedekind cut

Mathematics

A means of constructing the real numbers from the rational numbers, introduced by Richard Dedekind. A Dedekind cut is a subset A of ℚ such that:
1. (i) ∅ ≠ A ≠ ℚ.
2. (ii) if x < y are rational numbers and y ∈ A, then x ∈ A.
3. (iii) if x ∈ A, then there exists y ∈ A such that x < y.
  
  A Dedekind cut A then represents the real number that is its supremum but without having to make reference to real numbers. Order can be defined by A ≤ B if and only if A⊆B and addition by $A + B = {a + b : a \in A, b \in B} .$
  
  The other arithmetic operations can also be defined but are a little more involved as care must be taken with signs.

Philosophy

It has been known since the Greeks that there is no ratio of numbers, a/b, that is equal to the square root of 2. But there is no maximum ratio whose square is less than 2, and no minimum ratio whose square is greater than 2. The German mathematician Richard Dedekind in 1872 pointed out that each real number corresponds to a ‘cut’ like this in the class of ratios. This means that if we are given the set of rationals, we can construct reals in terms of sets of them: the set of rationals whose square is less than 2 is an open set that can represent the square root of 2. Mathematicians including Russell and Whitehead were thus able to identify a real with the class of all ratios less than it. This has the disadvantage that rational numbers are no longer a subset of the reals, but there are other ways of using Dedekind’s insight and preserving a uniform treatment for rational and irrational reals.

单词	Dedekind cut
释义	Dedekind cut Mathematics A means of constructing the real numbers from the rational numbers, introduced by Richard Dedekind. A Dedekind cut is a subset A of ℚ such that: (i) ∅ ≠ A ≠ ℚ. (ii) if x < y are rational numbers and y ∈ A, then x ∈ A. (iii) if x ∈ A, then there exists y ∈ A such that x < y. A Dedekind cut A then represents the real number that is its supremum but without having to make reference to real numbers. Order can be defined by A ≤ B if and only if A⊆B and addition by $A + B = {a + b : a \in A, b \in B} .$ The other arithmetic operations can also be defined but are a little more involved as care must be taken with signs. Philosophy It has been known since the Greeks that there is no ratio of numbers, a/b, that is equal to the square root of 2. But there is no maximum ratio whose square is less than 2, and no minimum ratio whose square is greater than 2. The German mathematician Richard Dedekind in 1872 pointed out that each real number corresponds to a ‘cut’ like this in the class of ratios. This means that if we are given the set of rationals, we can construct reals in terms of sets of them: the set of rationals whose square is less than 2 is an open set that can represent the square root of 2. Mathematicians including Russell and Whitehead were thus able to identify a real with the class of all ratios less than it. This has the disadvantage that rational numbers are no longer a subset of the reals, but there are other ways of using Dedekind’s insight and preserving a uniform treatment for rational and irrational reals.
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