This monograph is intended for Mathematics students that plan to continue their mathematical education at University, as well as students with interests in Analytic Philosophy or Theoretical Computer Science.Topics include Countable and Uncountable sets, Finite and Infinite sets, the sizes of Infinities, Countable Rational and Uncountable Real numbers, Power Set, Cantor’s theorem, Cantor’s Paradox, Russell’s paradox, Zermelo axioms for set theory, Essentials of Axiomatic method, Continuum Hypotheses, Unlimited Abstraction Principle and Separation Principle, Undecidability of Continuum Hypotheses in Zermelo-Fraenkel system, objections to Zermelo system, and other topics.Keywords: Axiomatic method, Dedekind cut, Complete Ordered Field, Cantor’s theorem, Continuum Hypotheses, Russell’s paradox, Zermelo-Fraenkel system, ZFC.