超实数系统与转移原则

科普中国介绍超实数，完全不提转移原则，是误人子弟。超实数传入中国，变味了。

在新的一年里，我们将有滋有味地引入超实数，奉献给国内广大读者，特别是，大学新生。              

我们深信，2019年将是超实数在中国普及的元年。

  请见本文附件，原汁原味地展现了超实数系统的原貌。。

袁萌  陈启清   12月30日

附件：超实数的“知识共享”版本

Hyperreals（注意：这是CC版本！）

Infinitesimals (ε) and infinites (ω) on the hyperreal number line (1/ε = ω/1)

The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

1 + 1 + + 1 {\displaystyle 1+1+\cdots +1}

 (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.[1]

The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic Law of Continuity. The transfer principle states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since

sin ( π n ) = 0 {\displaystyle \sin({\pi n})=0}

 for all integers n, one also has

sin ( π H ) = 0 {\displaystyle \sin({\pi H})=0}

 for all hyperintegers H. The transfer principle for ultrapowers is a consequence of o' theorem of 1955.

Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion.[2] In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated.

The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis. One immediate application is the definition of the basic concepts of analysis such as derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes

f ′ ( x ) = s t ( f ( x + Δ x ) − f ( x ) Δ x ) {\displaystyle f'(x)={\rm {st}}\left({\frac {f(x+\Delta x)-f(x)}{\Delta x}}\right)}

 for an infinitesimal

Δ x {\displaystyle \Delta x}

, where st(·) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.

Contents

1The transfer principle

2 Use in analysis

2.1 Calculus with algebraic functions

2.2 Integration

3 Properties

4 Development

4.1 From Leibniz to Robinson

4.2 The ultrapower construction

4.3 An intuitive approach to the ultrapower construction

5 Properties of infinitesimal and infinite numbers

6 Hyperreal fields

7 See also

8 References

9 Further reading

10 External links

The transfer principle

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