A clear, rigorous and thorough exposition of: *The Propositional Calculus*The Predicate Calculus *Set Theory*The different kinds of continuity and their respective natures*The different kinds of number: cardinal, ordinal, rational, real, complex, hyper-complex, and transfinite *The set-theoretic foundations of cardinal arithmetic and the various generalizations thereof*The nature of formal languages *Diagonalization and Incompleteness*The nature of logic and of formal truth, and *A proof of the incompleteness of any given system of deduction.