超实讲义

i foundations

1 what are the hyperreals

1.1 infinitely small and large

1.2 historical background

1.3 what is a real number

1.4 historical references

2 large sets

2.1 infinitesimals as variable quantities

2.2 largeness

2.3 filters

2.4 examples of filters

2.5 facts about filters

2.6 zorn's lemma

2.7 exercises on filters

3 ultrapower construction of the hyperreals

3.1 the ring of real-valued sequences

3.2 equivalence modulo an ultrafilter

3.3 exercises on almost-everywhere agreement

3.4 a suggestive logical notation

3.5 exercises on statement values

3.6 the ultrapower

3.7 including the reals in the hyperreals

3.8 infinitesimals and unlimited numbers

3.9 enlarging sets

3.10 exercises on enlargement

3.11 extending functions

3.12 exercises on extensions

3.13 partial functions and hypersequences

3.14 enlarging relations

3.15 exercises on enlarged relations

3.16 is the hyperreal system unique

4 the transfer principle

4.1 transforming statements

4.2 relational structures

4.3 the language of a relational structure

4.4 *-transforms

4.5 the transfer principle

4.6 justifying transfer

4.7 extending transfer

5 hyperreals great and small

5.1 (un)limited, infinitesimal, and appreciable numbers

5.2 arithmetic of hyperreals

5.3 on the use of "finite" and "infinite"

5.4 halos, galaxies, and real comparisons

5.5 exercises on halos and galaxies

5.6 shadows

5.7 exercises on infinite closeness

5.8 shadows and completeness

5.9 exercise on dedekind completeness

5.10 the hypernaturals

5.11 exercises on hyperintegers and primes

5.12 on the existence of infinitely many primes

ii basic analysis

6 convergence of sequences and series

6.1 convergence

6.2 monotone convergence

6.3 limits

6.4 boundedness and divergence

6.5 cauchy sequences

6.6 cluster points

6.7 exercises on limits and cluster points

6.8 limits superior and inferior

6.9 exercises on lim sup and lim inf

6.10 series

6.11 exercises on convergence of series

7 continuous functions

7.1 cauchy's account of continuity

7.2 continuity of the sine function

7.3 limits of functions

7.4 exercises on limits

7.5 the intermediate value theorem

7.6 the extreme value theorem

7.7 uniform continuity

7.8 exercises on uniform continuity

7.9 contraction mappings and fixed points

7.10 a first look at permanence

7.11 exercises on permanence of functions

7.12 sequences of functions

7.13 continuity of a uniform limit

7.14 continuity in the extended hypersequence

7.15 was cauchy right

8 differentiation

8.1 the derivative

8.2 increments and differentials

8.3 rules for derivatives

8.4 chain rule

8.5 critical point theorem

8.6 inverse function theorem

8.7 partial derivatives

8.8 exercises on partial derivatives

8.9 taylor series

8.10 incremental approximation by taylor's formula

8.11 extending the incremental equation

8.12 exercises on increments and derivatives

9 the riemann integral

9.1 riemann sums

9.2 the integral as the shadow of riemann sums

9.3 standard properties of the integral

9.4 differentiating the area function

9.5 exercise on average function values

10 topology of the reals

10.1 interior, closure, and limit points

10.2 open and closed sets

10.3 compactness

10.4 compactness and (uniform) continuity

10.5 topologies on the hyperreals

iii internal and external entities

11 internal and external sets

11.1 internal sets

11.2 algebra of internal sets

11.3 internal least number principle and induction

11.4 the overflow principle

11.5 internal order-completeness

11.6 external sets

11.7 defining internal sets

11.8 the underflow principle

11.9 internal sets and permanence

11.10 saturation of internal sets

11.11 saturation creates nonstandard entities

11.12 the size of an internal set

11.13 closure of the shadow of an internal set

11.14 interval topology and hyper-open sets

12 internal functions and hyperfinite sets

12.1 internal functions

12.2 exercises on properties of internal functions

12.3 hyperfinite sets

12.4 exercises on hyperfiniteness

12.5 counting a hyperfinite set

12.6 hyperfinite pigeonhole principle

12.7 integrals as hyperflnite sums

iv nonstandard frameworks

13 universes and frameworks

13.1 what do we need in the mathematical world

13.2 pairs are enough

13.3 actually, sets are enough

13.4 strong transitivity

13.5 universes

13.6 superstructures

13.7 the language of a universe

13.8 nonstandard frameworks

13.9 standard entities

13.10 internal entities

13.11 closure properties of internal sets

13.12 transformed power sets

13.13 exercises on internal sets and functions

13.14 external images are external

13.15 internal set definition principle

13.16 internal function definition principle

13.17 hyperfiniteness

13.18 exercises on hyperfinite sets and sizes

13.19 hyperfinite summation

13.20 exercises on hyperfinite sums

14 the existence of nonstandard entities

14.1 enlargements

14.2 concurrence and hyperfinite approximation

14.3 enlargements as ultrapowers

14.4 exercises on the ultrapower construction

15 permanence, comprehensiveness, saturation

15.1 permanence principles

15.2 robinson's sequential lemma

15.3 uniformly converging sequences of functions

15.4 comprehensiveness

15.5 saturation

v applications

16 loeb measure

16.1 rings and algebras

16.2 measures

16.3 outer measures

16.4 lebesgue measure

16.5 loeb measures

16.6 μ-approximability

16.7 loeb measure as approximability

16.8 lebesgue measure via loeb measure

17 ramsey theory

17.1 colourings and monochromatic sets

17.2 a nonstandard approach

17.3 proving p, amsey's theorem

17.4 the finite ramsey theorem

17.5 the paris-harrington version

17.6 reference

18 completion by enlargement

18.1 completing the rationals

18.2 metric space completion

18.3 nonstandard hulls

18.4 p-adic integers

18.5 p-adic numbers

18.6 power series

18.7 hyperfinite expansions in base p

18.8 exercises

19 hyperfinite approximation

19.1 colourings and graphs

19.2 boolean algebras

19.3 atomic algebras

19.4 hyperfinite approximating algebras

19.5 exercises on generation of algebras

19.6 connecting with the stone representation

19.7 exercises on filters and lattices

19.8 hyperfinite-dimensional vector spaces

19.9 exercises on (hyper) real suhspaces

19.10 the hahn-banach theorem

19.11 exercises on (hyper) linear functionals

20 books on nonstandard analysis

index

《超实讲义》是由世界图书出版公司出版的。《超实讲义》是一部讲述非标准分析的入门教程，是由作者数年教学讲义发展并扩充而成。具备基本分析知识的高年级本科生，研究生以及自学人员都可以完全读懂。非标准分析理论不仅是研究无限大和无限小的强有力的理论，也是一种截然不同于标准数学概念和结构的方法，更是新的结构，目标和证明的源泉，推理原理的新起点。书中是从超实数系统开始，从非标准的角度讲述单变量积分，分析和拓扑，着重强调变换原理作为一个重要的数学工具的重要作用。数学宇宙的讲述为全面研究非标准方法论提供了基础保证。最后一章着眼于应用，将这些理论应用于loeb 测度理论及其与lebesgue 的一些关系，ramsey 定理，p-进数的非标准结构和幂级数，boolean 代数的stone 表示定理的非标准证明和hahn-banach 定理。《超实讲义：英文(影印版)》的最大特点尽早引入内集，外集，超有限集，以及集理论扩展方法，较常规的建立在超结构基础上，这样的方式更加显而易见。读者对象：数学专业的高年级本科生，研究生和科研人员。【作者简介】作者：（新西兰）哥德布拉特（Robert Goldblatt）