In a book about logic (propositional logic), author uses sets to describe propositional variables, operators (eg. $A=\lbrace \neg,\wedge,\vee,\Rightarrow,\Leftrightarrow\rbrace$)... How can those sets even exist? What are those operators exactly? (obviously not sets, althought in axiomatic set theory everything is set, so does it means that sets such as $A$ does not exist? What is it then, just a proper class?)
So, in other words, are mathematical sets and sets in logic different? (Mathematical sets made by axioms, and sets in logic made just intuitively?)
Another thing which I would like to see is a step by step construction of all basic mathematics such that you start by nothing. In other words, no use of anything which hasn't yet been defined. (e.g., if you start with logic, then you cant use sets in logic, or functions, or numbers because you haven't yet even defined them). So, is there some book written in that way, or maybe some internet resource?