Elements of Logic via Numbers and Sets (Springer Undergraduate Mathematics Series)
From the reviews: “It took me a while to appreciate the need for a course intended to introduce mathematics undergraduates to advanced mathematics after they finish the calculus sequence. … This was the first book I found that seems really close to the core. It is well-written and very well suited to such a course.” (James M. Cargal, UMAP Journal, Vol. 31 (1), 2010)