In Kelley book on topology, in the appendix on elementary set theory, he says in the second paragraph, that "a working knowledge of elementary logic is assumed, but acquaintance with formal logic is not essential. However, an understanding of the nature of a mathematical system (in the technical sense) helps to clarify and motivate some of the discussion. Tarski's excellent exposition [here he refers to Tarski's "Introduction to Logic"] describes such system very lucidly and is particularly recommended for general background."
My questions are:
1) What is the difference between elementary logic and formal logic ? Shall I interpret "elementary logic" as those mental processes that enable me do to logic reasoning and inference (for when I deal with strings of symbols that have a mathematical meaning - as when I do when for example I would work with the axioms of ZFC to derive results) ?
2) What is a "mathematical system" ? (I presently don't have the means to look up Tarski's book, to see what Tarski himself wrote there, what it is that Kelley describes as a "mathematical system")