I have been partially inspired by questions here to attempt to create a good axiomatization for the integers, the real numbers, and the complex numbers. Anyone interested can find the result of my efforts here: https://www.dropbox.com/s/6rfknwn28kr0cv5/AxiomList.pdf?dl=0 (8 pages)
I have used as much as I can from the book "Introduction to Logic and the Methodology of the deductive Sciences" by Alfred Tarski in my attempt.
The text begins with some notes on mathematical logic, and continues with some notes on: groups, rings, and fields. There is also some notes on operator properties used as generalizations.
When I go into the Integers, Reals, and Complex numbers, I have made an attempt to use the operator properties to reduce the number of necessary axioms. The axioms for the Integers began as a copy of axioms from a math book. [Biggs: Discrete Mathematics, if I remember correctly.] Axioms for groups, rings, and fields have the same source. I attempted to translate them into predicate logic, since they were in a non-formal language.
My work with Tarskis axiomatization of the reals prompted some more of my editing, and the proposed axioms for the complex numbers are entirely my own. (I have not been able to find any set of axioms for the complex nubers on Internet. Might be there, but I have not found them.)
I am educated in astronomy, physics, and computer science, so math is not entirely new to me. However, I am relatively new to predicate logic, and might have made errors due to my inexperience.
I would therefore like to make a request for comments (RFC), on my logic statements to this group. I am willing to correct any errors and add any suggested improvements. If anyone would like to use it, or an improved version, be my guest.
Thanks in advance for any comments or suggestions.
Edit: Found a two improvements in my axiomatization of the complex numbers, and added some text about the axioms in order to clarify.
Edit 2: attempted to make the header more relevant. Added more information on what is my contribution and not.