A rigorous treatment of the hyperreal numbers and nonstandard analysis.

Question

I tried finding a decent book to study nonstandard analysis from and found Goldblatt's Lectures on the Hyperreals. However, I was very disappointed to find out that the text is not rigorous at all --- Goldblatt doesn't even prove the transfer principle. Are there any completely rigorous treatments of the hyperreals?

Ideally, I would like them to cover the same content that Goldblatt's book does: construction, nonstandard analysis and miscellaneous applications. However, I'd be more than satisfied with just construction and nonstandard analysis. I would also like the book to be more-or-less self-contained in this regard. That is, if the book simply states the transfer principle is a consequence of some stronger model-theoretic theorem, I don't gain anything from that. Of course, some knowledge of analysis, algebra and logic must be assumed, and I'm fine with all of those on a basic level.

To clarify, I am also not interested in axiomatic approaches to these structures. So, for example, a construction of the hyperreals would be much preferred over an axiomatic introduction.

Answer 1

The original 1966 classic text is:

Robinson, Abraham, Non-standard analysis, Princeton, NJ: Princeton Univ. Press. xix, 293 p. (1996). ZBL0843.26012.

This is a reprint of the 1974 second edition.

Answer 2

Nonstandard Analysis: Theory and Applications (1997) edited by Arkeryd, Cutland, and Henson.

Covers "simple" nonstandard analysis (e.g. nonstandard extensions of a "simple" set), then "higher-order" nonstandard exstensions culminating in extensions of superstructures, and then goes on to cover real analysis, topology, probability, ODEs, etc.