Hyperreals and non-standard analysis

The construction of the hyperrreals amounts to “supplement” the usual real numbers with infinitesimals and infinities. This has to be done in a such a way that some key properties of the real numbers are preserved (i.e. for all real numbers a,b it is true that a+b=b+a, and this will still be true for all hyperreal numbers). On the other hand some properties that are true for the real numbers will not be true for hyperreal numbers (i.e. there exist a (hyperreal) number smaller than all positive real numbers and larger than 0!!).

In my opinion this project should address these key points:

1. Understand what properties are preserved, and which are neglected when defining the hyperreals and why. This is the idea behind the “transfer principle”.
2. Learn how to perform basic operations with infinitesimals, without making mistakes.
3. How the usual problems in calculus are solved with infinitesimals. How is continuity defined? and a limit? and a derivative?
4. Ideally you will have an understanding on how the hyperreal numbers are rigorously defined. Usually hyperreal numbers are defined via sequences.

All these points are beautifully explained in the book by H. Jerome Keisler, freely available from the web. In particular the key sections are 1.4-1.6, chapter 2, sections 3.1-3.4, 5.8 and finally the Epilogue, where you will find the real “formal” definition of the hyperreal numbers. This last point is also very well explained in this mathematical essay.