非标准模型理论的若干问题研究
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摘要
本文讨论了滤子、超结构和超幂模型及其性质,并分解*-映射为M-映射和e-映射.讨论了全域和形式语言,进而用M-映射和e-映射的具体构造证明了转换原理.在此基础上,讨论了个体集S,指标集I和滤子F三个因素对非标准模型的影响,并给出了非标准扩大模型和饱和模型的具体构造.最后讨论了非标准扩大模型和饱和模型成立的一些充要条件以及非标准扩大模型和饱和模型的关系.
论文的要点及主要内容如下：
在第一章和第二章里,首先简单概述了非标准分析产生背景、发展状况及其研究现状,其次给出了滤子的定义和几种常见的超滤子及其性质,最后给出了它们之间的关系.
第三章,首先讨论了超结构的定义及其性质,其次介绍了超幂模型,最后给出了映射M和e,及其有关性质和结论,并得出了M-映射和e-映射合成*-映射,即得出了*-映射的内部构造.
第四章,讨论了全域和形式语言,在此基础上,利用第三章给出*-映射分解成的M-映射和e-映射,证明了转换原理.
在第五章中,通过讨论个体集S,指标集I和滤子厂三个因素对非标准模型的影响,进而定义了良的非标准模型,在此基础上,讨论了当个体集S,指标集I,滤子F分别满足什么条件时,非标准模型才是非标准扩大模型或饱和模型.最后讨论了非标准扩大模型和饱和模型成立的一些充要条件和非标准扩大模型和饱和模型的关系.
In this paper, the filter, the superstructure and the superpower model are dis-cussed. Then some properties of them are presented. The natural construction of *-mapping, the universe and languages are discussed. Based on it, the transfer principle is showed by using mapping M and e. Then three effective factors of the nonstandard model,that is, the individuals S, index set I and the filter F are discussed. The detailed construction of the nonstandard enlarged model and the saturated model are given. Finally, the sufficient and necessary conditions of the nonstandard enlarged model and the saturated model are showed, and the relation of the nonstandard enlarged model and the saturated is proved.
The main points of this paper are as follows:
In the first two chapters, the background, the actual development condition and research states of the nonstandard analysis are introduced. Then definition of the filter, some other ultrafilters and properties of the ultrafilter are presented. Finally, the relation of these ultrafilters are given.
In the third chapter, the definition and properties of the superstructure are in-troduced. The mapping M, mapping e and the related properties and conclusions are presented. Then it is proved that the mapping M and the mapping e compose the *-mapping, and the natural construction of *-mapping is given.
In the fourth chapter, the universe and the languages are disscused. Based on it, the transfer principle is proved by using mapping M and mapping e.
In the fifth chapter,the three effective factors of the nonstandard model,that is, the individuals S,index set I and the filter F are discussed. Then the well nonstandard model is defined. Based on it, the construction of the nonstandard enlarged model and the saturated model are givn by discussing the individuals S, the index set I and the fil-ter F. Then the sufficient and necessary conditions of the nonstandard enlarged model and the saturated model are showed, and the relation of the nonstandard enlarged model and the saturated is proved.
论文的要点及主要内容如下：
在第一章和第二章里,首先简单概述了非标准分析产生背景、发展状况及其研究现状,其次给出了滤子的定义和几种常见的超滤子及其性质,最后给出了它们之间的关系.
第三章,首先讨论了超结构的定义及其性质,其次介绍了超幂模型,最后给出了映射M和e,及其有关性质和结论,并得出了M-映射和e-映射合成*-映射,即得出了*-映射的内部构造.
第四章,讨论了全域和形式语言,在此基础上,利用第三章给出*-映射分解成的M-映射和e-映射,证明了转换原理.
在第五章中,通过讨论个体集S,指标集I和滤子厂三个因素对非标准模型的影响,进而定义了良的非标准模型,在此基础上,讨论了当个体集S,指标集I,滤子F分别满足什么条件时,非标准模型才是非标准扩大模型或饱和模型.最后讨论了非标准扩大模型和饱和模型成立的一些充要条件和非标准扩大模型和饱和模型的关系.
In this paper, the filter, the superstructure and the superpower model are dis-cussed. Then some properties of them are presented. The natural construction of *-mapping, the universe and languages are discussed. Based on it, the transfer principle is showed by using mapping M and e. Then three effective factors of the nonstandard model,that is, the individuals S, index set I and the filter F are discussed. The detailed construction of the nonstandard enlarged model and the saturated model are given. Finally, the sufficient and necessary conditions of the nonstandard enlarged model and the saturated model are showed, and the relation of the nonstandard enlarged model and the saturated is proved.
The main points of this paper are as follows:
In the first two chapters, the background, the actual development condition and research states of the nonstandard analysis are introduced. Then definition of the filter, some other ultrafilters and properties of the ultrafilter are presented. Finally, the relation of these ultrafilters are given.
In the third chapter, the definition and properties of the superstructure are in-troduced. The mapping M, mapping e and the related properties and conclusions are presented. Then it is proved that the mapping M and the mapping e compose the *-mapping, and the natural construction of *-mapping is given.
In the fourth chapter, the universe and the languages are disscused. Based on it, the transfer principle is proved by using mapping M and mapping e.
In the fifth chapter,the three effective factors of the nonstandard model,that is, the individuals S,index set I and the filter F are discussed. Then the well nonstandard model is defined. Based on it, the construction of the nonstandard enlarged model and the saturated model are givn by discussing the individuals S, the index set I and the fil-ter F. Then the sufficient and necessary conditions of the nonstandard enlarged model and the saturated model are showed, and the relation of the nonstandard enlarged model and the saturated is proved.
引文
[1]冯汉桥.超实数域*R的基数[J].陕西理工学院学报(自然科学版),1990,18(1)：11-13.
[2]陈东立.关于超实数域*R的基数的一些注记[J].商丘师范学院学报,2001,17(4)：56-60.
[3]张福泰,冯汉桥.The Topological Structures of the Hyperreal Number Field *R[J].数学进展,1997,26(5)：435-439.
[4]张福泰.内超实度量空间中的拓扑[J].陕西师范大学学报(自然科学版),1997,25(2)：14-16.
[5]A.E.Hurd, P.A.Loeb..An Introduction to Nonstandard Real Analysis [M].New York: Academic Press,1985:7.
[6]Cutland.N.J..Nonstandard Measure Theory and Its Application [J]Bull. London Math.Soc,1983,15,529-589.
[7]A.Robinson. Nonstandard Analysis[M].Amsterdam:North-holland,1966:25.
[8]陈东立. The Extension of Infinitesimal Prolongation Theorem and Its Applica-tion [J]数学研究与评论,2003,23(2)：221-224.
[9]陈东立,马春晖,史艳维.The Structure of *τx and Its Properties [J].数学研究与评论,2007,27(4)：671-673.
[10]陈东立,马春晖,史艳维.The Extension of Robinson's Sequential Lemma and Its Application [J]数学季刊,2007,22(3)：471-474.
[11]陈东立,马春晖,苑文法.滤子收敛的非标准特征及其应用[J].纯粹数学与应用数学,2004,20(1)：10-13.
[12]陈东立.Radon空间的非标准表示[J].数学研究与评论,2002,22(3)：493-496.
[13]陈东立,史艳维,马春晖.内涵数逼近定理及上溢原理的推广及应用[J].纯粹数学与应用数学,2006,22(1)：32-36.
[14]陈东立,马春晖,史艳维.扩大模型的充要条件及其应用[J].西安建筑科技大学学报,2004,36(2)：441-443.
[15]潘红霞,陈东立,史艳维.饱和模型的充要条件及其应用[J].纯粹数学与应用,2006,22(1)：131-133.
[16]Kelley J.L. General Topology[M]. New York:Sringer Verlag,1955:37.
[17]D. Scott. Measurable Cardinals and Constructible Sets[J]. Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys.,1961,9:521-524.
[18]S. Ulam. Zur Mass Theorie Der Allgemeinen Mengenlehre[J]. Fundamenta Math, 1930,16:140-150.
[19]李邦河.非标准分析基础[M].上海：科技出版社,1986：29.
[20]Davis M. Applied Nonstandard Analysis[M]. New York:Wiley,1977:16.
[21]D.Grainger. Flat Set[J]. Journal of Symbolic Logic,1994,59(3):1012-1021.
[22]H.J.Keisler. Ultraproducts and Saturated Models[J].Indag.Math.,1964,26:179-186.
[23]H.J.Keisler. Good Ideals in Fields of Sets[J]. Ann. Math.,1964,79:338-359.
[24]Luxem burg W.A.J.. General Theory of Monads[M]. New York:Halt,1969:39.
[25]Bernstein. A. R..Nonstandard Analysis in Studies in Model Theory[J]. MAA Studies in Math.,1973,8:35-38.
[26]Sack.G..Saturated Model Theory[M]. Reading, Mass, Benjamin,1972:11.
[27]N.J.Cutland (ed.).Nonstandard Analysis and Its Applications[M]. Cam-bridge:Cambridge University Press,1988:87.
[28]Chang.C.C., Keisler,H.J.. Model Theory[M]. North Holland:Amsterdam press, 1973:15.
[29]Kamo,S.. Nonstandard Natural Number Systems and Nonstandard Model[J]. to Appear in Journal of Symbolic Logic.
[30]H.J.Keisler. Ultraproducts Which Are Not Saturated[J]. Journal of Symbolic Logic,1967,32:23-46.
[31]张卫国,韩俊峰.S-饱和集的性质及其应用[J].西安科技大学学报,2007,27(3)：520-522.
[32]E.Nelson. Internal Set Theory[J]. Bull. Amer. Math. Soc.,1977,8(3):1165-1193.
[33]K.D.Stroyan, W.A. J.Luxemburg. Introduction to the Theory of Infinitesimals [M]. New York: Academic Press,1976:49.
[34]S.Salbany, T.Todorov. Monads and Realcompactnes[J]. Topology and Its Applica-tions,1994,56:99-104.
[35]C.W.Henson. Analystic Sets, Baire Sets and the Standard Part Map [J]. Can. J. Math.,1979,31:663-672.
[36]Anderson R M. Star-finite Representations of Measure Spaces[J]. Trans. Amer. Math. Soc,1982,271:667-687.
[2]陈东立.关于超实数域*R的基数的一些注记[J].商丘师范学院学报,2001,17(4)：56-60.
[3]张福泰,冯汉桥.The Topological Structures of the Hyperreal Number Field *R[J].数学进展,1997,26(5)：435-439.
[4]张福泰.内超实度量空间中的拓扑[J].陕西师范大学学报(自然科学版),1997,25(2)：14-16.
[5]A.E.Hurd, P.A.Loeb..An Introduction to Nonstandard Real Analysis [M].New York: Academic Press,1985:7.
[6]Cutland.N.J..Nonstandard Measure Theory and Its Application [J]Bull. London Math.Soc,1983,15,529-589.
[7]A.Robinson. Nonstandard Analysis[M].Amsterdam:North-holland,1966:25.
[8]陈东立. The Extension of Infinitesimal Prolongation Theorem and Its Applica-tion [J]数学研究与评论,2003,23(2)：221-224.
[9]陈东立,马春晖,史艳维.The Structure of *τx and Its Properties [J].数学研究与评论,2007,27(4)：671-673.
[10]陈东立,马春晖,史艳维.The Extension of Robinson's Sequential Lemma and Its Application [J]数学季刊,2007,22(3)：471-474.
[11]陈东立,马春晖,苑文法.滤子收敛的非标准特征及其应用[J].纯粹数学与应用数学,2004,20(1)：10-13.
[12]陈东立.Radon空间的非标准表示[J].数学研究与评论,2002,22(3)：493-496.
[13]陈东立,史艳维,马春晖.内涵数逼近定理及上溢原理的推广及应用[J].纯粹数学与应用数学,2006,22(1)：32-36.
[14]陈东立,马春晖,史艳维.扩大模型的充要条件及其应用[J].西安建筑科技大学学报,2004,36(2)：441-443.
[15]潘红霞,陈东立,史艳维.饱和模型的充要条件及其应用[J].纯粹数学与应用,2006,22(1)：131-133.
[16]Kelley J.L. General Topology[M]. New York:Sringer Verlag,1955:37.
[17]D. Scott. Measurable Cardinals and Constructible Sets[J]. Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys.,1961,9:521-524.
[18]S. Ulam. Zur Mass Theorie Der Allgemeinen Mengenlehre[J]. Fundamenta Math, 1930,16:140-150.
[19]李邦河.非标准分析基础[M].上海：科技出版社,1986：29.
[20]Davis M. Applied Nonstandard Analysis[M]. New York:Wiley,1977:16.
[21]D.Grainger. Flat Set[J]. Journal of Symbolic Logic,1994,59(3):1012-1021.
[22]H.J.Keisler. Ultraproducts and Saturated Models[J].Indag.Math.,1964,26:179-186.
[23]H.J.Keisler. Good Ideals in Fields of Sets[J]. Ann. Math.,1964,79:338-359.
[24]Luxem burg W.A.J.. General Theory of Monads[M]. New York:Halt,1969:39.
[25]Bernstein. A. R..Nonstandard Analysis in Studies in Model Theory[J]. MAA Studies in Math.,1973,8:35-38.
[26]Sack.G..Saturated Model Theory[M]. Reading, Mass, Benjamin,1972:11.
[27]N.J.Cutland (ed.).Nonstandard Analysis and Its Applications[M]. Cam-bridge:Cambridge University Press,1988:87.
[28]Chang.C.C., Keisler,H.J.. Model Theory[M]. North Holland:Amsterdam press, 1973:15.
[29]Kamo,S.. Nonstandard Natural Number Systems and Nonstandard Model[J]. to Appear in Journal of Symbolic Logic.
[30]H.J.Keisler. Ultraproducts Which Are Not Saturated[J]. Journal of Symbolic Logic,1967,32:23-46.
[31]张卫国,韩俊峰.S-饱和集的性质及其应用[J].西安科技大学学报,2007,27(3)：520-522.
[32]E.Nelson. Internal Set Theory[J]. Bull. Amer. Math. Soc.,1977,8(3):1165-1193.
[33]K.D.Stroyan, W.A. J.Luxemburg. Introduction to the Theory of Infinitesimals [M]. New York: Academic Press,1976:49.
[34]S.Salbany, T.Todorov. Monads and Realcompactnes[J]. Topology and Its Applica-tions,1994,56:99-104.
[35]C.W.Henson. Analystic Sets, Baire Sets and the Standard Part Map [J]. Can. J. Math.,1979,31:663-672.
[36]Anderson R M. Star-finite Representations of Measure Spaces[J]. Trans. Amer. Math. Soc,1982,271:667-687.