Book on the rigorous construction of logic and set theory

Is there a book out there that discusses the construction of logic and set theory from the ground up?

Many of the logic books that I've found suffers from one main problem: it doesn't define things in the right order. For example:

1- propositions are statements which can either be true or false

2- predicates are ... (has references to propositions)

3- quantifier are ... (has references to predicates and propositions)

4- an axiom system contains a finite number of ... countable set of ... etc.

So the problem becomes: what does 'finite' mean? what is a 'number'? what is a 'countable'? have you even defined what a 'set' is?

In other words, it goes from step 1, step 2, step 3, refers to step 513, step 4, step 5, ...

I am aware of these concepts beforehand, but it would be nice to know the proper sequence of definitions (instead of defining something in terms of other things that have yet to be defined).

• An exposition having a step by step construction and logical building up concepts and that covers most parts of what you have asked, is the book named "The Topology Problem Solver" by Emil G. Milewski. Here is the link There might certainly be many other such expositions, but this came to my mind first. – spkakkar Mar 18 '18 at 7:57
• Some hints: G.Tourlakis, Lectures in Logic and Set Theory: Volume 1, Mathematical Logic and Volume 2 and W.Felscher, Lectures on Mathematical Logic Volume I-III. – Mauro ALLEGRANZA Mar 18 '18 at 9:42
• Of possible interest is my answer to the Mathematics Stack Exchange question Foundation of Formal Logic. – Dave L. Renfro Mar 18 '18 at 11:13
• You might think of such statements as informal commentary to aid in getting "the big picture." Like comments in computer programming source code. You can say, for example, that ZFC has "countably many" axioms in a presentation of the axioms, but you don't use this notion when formally proving theorems in ZFC. – Dan Christensen Mar 18 '18 at 14:47
• @MauroALLEGRANZA could you please post an answer with the Tourlakis sources so that I could accept it. I found a copy and It's definitely what I'm looking for. Thanks! – japseow Apr 4 '18 at 4:11