The Exposition Of Some Questions Of The Nonstandard Model Theory

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.