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).