# book recommendation - formal systems

I'm looking for a strict book/pdf about logic which discusses formal systems in great detail. I only know basic stuff. It should cover:

• definitions (like $(\exists x \varphi\leftrightarrow\lnot\forall x\lnot\varphi))$ inside the formal system (what to take care of when making definitions in a formal system)

• distinct variables (and that we can make a formal system without the concept of "free" variables)

• substitutions (would be nice if most would be taken care of "inside" the formal system)

• using classes in zfc in a way, that they can always be eliminated (with proof)

• defining ordinal addition etc. "inside" the formal system and working with the definition "inside" the formal system.

I shall suggest the resources linked from here. In particular since you seem to know basic logic, you probably should go straight to Stephen Simpson's notes. As for your desire to use classes in ZFC in a way that can be eliminated, you can rigorously manipulate definable classes and class functions by adding definitorial expansion to ZFC, because it is conservative and can be eliminated, which any proper logic textbook should mention (such as Rautenberg's in my first link). I think my second link should also address your question about ordinals; they form a definable class over ZFC and hence you can formally reason about them via definitorial expansion.

• Thanks for the links! But this isn't really what I'm looking for. I'm looking for a book/pdf which actually is creating a fromal system and is working in it (as rigorous as possible, ofcourse it can't contain a formal proof for everything but there should be some in it). It should also at some point define the ordinals in the system (and addition, multiplication... of them). Your answer about definitorial expansion looks interesting, I will look into it! – owo Apr 7 '17 at 12:09
• @owo: It is hard to find such a book because most mathematicians and logicians are comfortable working with natural language, as long as they can be sure what they do can be translated into a suitable formal system, usually ZFC for modern mathematics. I know you're looking for a more explicitly formal kind, but actually as long as you fully grasp the notion of definitorial expansion, you will be able to mentally formalize what you read. – user21820 Apr 8 '17 at 7:43
• @owo: I also realize that I missed one point in your question about having no free variables. I personally prefer that style, and if you're interested you can look up my description of one Fitch-style system that enforces every variable to be bound and also no variable shadowing. I don't know of any book with proofs written in that kind of Fitch-style format though. – user21820 Apr 8 '17 at 7:46