This is cross posted from [MathStackExchange][1]. Since this is a reference request, I believe there will not be duplications of efforts in answers. This is also related to the question [here][2]. 


In textbooks on logic I see a tradition of using notions of *set* and *map* from the very beginning of the exposition, long before the set theory is formally constructed. This causes multiple misunderstandings for the reader (I remember these difficulties when I was a student, and even now I have questions, as you can see), since formally this is a violation of the principle that a mathematician can't use a term before giving its accurate definition. That is why I believe that there must be texts where this mistake is repaired. Can anybody advise me a textbook on logic with a "linear structure", without these "circles in definitions"?

I already told this to people at MSE, I don't see serious technical obstacles for such a book to exist: the author can just formulate the axioms of predicate calculus + axioms of set theory, then consider the corollaries (i.e. construct a first-order theory of sets), and only after that discuss what they call "semantics of logic" (i.e. everything connected to interpretations of first order languages)? That is strange, I can't find such a textbook. I would appreciate very much if somebody could give a reference. 

P.S. I have a feeling that I must specify the question, because people seem to do not quite understand what I ask. Look at these examples.

1. Elliott Mendelson in his [Introduction to Mathematical Logic][3] formulates and proves in Chapter 2 (Corollary 2.14) [the Gödel completeness theorem][4]. Of course, for formulating and proving this he needs the notion of logically valid formula. He gives this definition a little bit earlier (also in Chapter 2), and of course, he uses the notion of interpretation, which he defines *with the help of the notions of set and function*. These are not sets and functions in some "trivial, everyday sense" -- these are "true sets" and "true functions" in the sense of Set theory. But he builds Set theory (and gives accurate definitions of sets and functions) only in Chapter 4, long after Gödel's theorem. 

2. The same happens in the book by Joseph R. Shoenfield [Mathematical logic][5]. He also defines valid formulas, *with the help of sets and functions*, and also proves Gödel's completeness theorem (in his Chapter 4), and he does this long before constructing Set theory (which appears only in Chapter 9). 

3. And this is everywhere, in all the textbooks on Logic that I know. The only difference is that sometimes the author does not build Set theory at all (like P.S.Novikov in his [Elements of Mathematical Logic][6]). 

So my question is

> Is there a book on logic where sets and functions are mentioned only after constructing ("true", axiomatic) Set theory?  

I understand that the word "set" can be used in non-mathematical, everyday sense, but as I wrote above, *Gödel completeness theorem is not that case*. I also understand the desire to have a similar notion for simpler, "everyday" situations, say, for describing syntax of your language. But this desire does not imply the necessity to use the word "set", which appears in your text later as a term of a rigoruos, axiomatic theory (where it has a much more sophisticated meaning than in everyday life). The English language is rich enough (as well as other languages, Russian, French, etc.), it is always possible to find a better solution, which does not provoke misunderstandings, accusations and controversy. Finally, it is not nice to arrange the course of a mathematical discipline in that way, because students (and readers of your book) perceive this as a mockery.  

A natural way to overcome this, I believe, is what I wrote before: 

> to construct a first-order theory of sets (with all axioms, including the axioms of predicate logic), and only after that to speak about things like "interpretation", "completeness", etc. 

Is it possible that nobody did this up to now?

P.P.S. To people who vote to close: I hope you have convincing explanations of your motives.

  [1]: https://math.stackexchange.com/questions/1946976/are-there-textbooks-on-logic-where-the-references-to-set-theory-appear-only-afte
  [2]: https://math.stackexchange.com/questions/173735/how-to-avoid-perceived-circularity-when-defining-a-formal-language
  [3]: https://www.amazon.com/Introduction-Mathematical-Discrete-Mathematics-Applications/dp/1584888768
  [4]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem
  [5]: https://www.amazon.com/Mathematical-Logic-Addison-Wesley-Joseph-Shoenfield/dp/1568811357
  [6]: https://www.amazon.com/Elements-Mathematical-Logic-P-S-NOVIKOV/dp/B0000CM3ZT