超实讲义（英文影印版）
出版时间：2011年版
内容简介
i foundations1 what are the hyperreals？1.1 infinitely small and large1.2 historical background1.3 what is a real number？1.4 historical references2 large sets2.1 infinitesimals as variable quantities2.2 largeness2.3 filters2.4 examples of filters2.5 facts about filters2.6 zorn's lemma2.7 exercises on filters3 ultrapower construction of the hyperreals3.1 the ring of real-valued sequences3.2 equivalence modulo an ultrafilter3.3 exercises on almost-everywhereagreement3.4 a suggestive logical notation3.5 exercises on statement values3.6 the ultrapower3.7 including the reals in the hyperreals3.8 infinitesimals and unlimited numbers3.9 enlarging sets3.10 exercises on enlargement3.11 extending functions3.12 exercises on extensions3.13 partial functions and hypersequences3.14 enlarging relations3.15 exercises on enlarged relations3.16 is the hyperreal system unique？4 the transfer principle4.1 transforming statements4.2 relational structures4.3 the language of a relational structure4.4 *-transforms4.5 the transfer principle4.6 justifying transfer4.7 extending transfer5 hyperreals great and small5.1 （un）limited， infinitesimal， and appreciablenumbers5.2 arithmetic of hyperreals5.3 on the use of \finite\ and \infinite\5.4 halos， galaxies， and real comparisons5.5 exercises on halos and galaxies5.6 shadows5.7 exercises on infinite closeness5.8 shadows and completeness5.9 exercise on dedekind completeness5.10 the hypernaturals5.11 exercises on hyperintegers and primes5.12 on the existence of infinitely manyprimesii basic analysis6 convergence of sequences and series6.1 convergence6.2 monotone convergence6.3 limits6.4 boundedness and divergence6.5 cauchy sequences6.6 cluster points6.7 exercises on limits and cluster points6.8 limits superior and inferior6.9 exercises on lim sup and lim inf6.10 series6.11 exercises on convergence of series7 continuous functions7.1 cauchy's account of continuity7.2 continuity of the sine function7.3 limits of functions7.4 exercises on limits7.5 the intermediate value theorem7.6 the extreme value theorem7.7 uniform continuity7.8 exercises on uniform continuity7.9 contraction mappings and fixed points7.10 a first look at permanence7.11 exercises on permanence of functions7.12 sequences of functions7.13 continuity of a uniform limit7.14 continuity in the extendedhypersequence7.15 was cauchy right？8 differentiation8.1 the derivative8.2 increments and differentials8.3 rules for derivatives8.4 chain rule8.5 critical point theorem8.6 inverse function theorem8.7 partial derivatives8.8 exercises on partial derivatives8.9 taylor series8.10 incremental approximation by taylor'sformula8.11 extending the incremental equation8.12 exercises on increments andderivatives9 the riemann integral9.1 riemann sums9.2 the integral as the shadow of riemannsums9.3 standard properties of the integral9.4 differentiating the area function9.5 exercise on average function values10 topology of the reals10.1 interior， closure， and limit points10.2 open and closed sets10.3 compactness10.4 compactness and （uniform） continuity10.5 topologies on the hyperrealsiii internal and external entitiesiv nonstandard frameworksv applicationsindex
目录
1foundations
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
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
