Looking for derivation of "axioms" for the real numbers from ZFC / Peano Axioms (1978?)

Question

I am looking for a derivation of the "axioms" for the real numbers starting from scratch with ZFC / Peano Axioms. I don't recall the title or author, but the publication date may have been around 1978.

EDIT: Any derivation would do. I had just seen a passing reference to such a work some time ago, one that was published in 1978. I want to use my DC Proof software to start from Peano's Axioms and use set theory to formally construct R, then prove the field axioms, etc. I know this is a huge project. I tried several years ago, but got bogged on even elementary proofs of the properties of Dedekind cuts. I am thinking of giving it another try, with some professional help this time.

Answer 1

I'm not sure if you're asking for any reference that provides this or a specific one. If I recall correctly, Terry Tao's analysis book at least does something close to this. He doesn't really construct the natural numbers in ZFC, but he builds the real numbers assuming the natural numbers exist satisfying the Peano axioms. It's also a very good intro to analysis.

Answer 2

Halmos ,P ,Naive set theory then Landau ,Foundations of Analysis will get the whole construction. Mendelson ,Elliot ,Number Systems and the Foundations of Analysis (Dover) has it all except for the derivation of the Piano axioms from set theory (see Halmos for this ) and more -this is an excellent book . Also Rudin -Principles of Mathematical Analysis has most of the Dedekind Cut construction except that he left out the proofs for the multiplication of reals ;Mendelson has this In the appendix -it is easy to overlook this.