 Reverse mathematics uses \(Z_2\) to isolate the exact axioms on which well-known theorems from mathematics rely. In a textbook such theorems are proved perhaps in an informal set theory, but how much set theory is actually needed in each case? For example, we may ask what is the weakest set of axioms from which the Bolzano-Weierstrass  Theorem[21]  can be proved? How much set theory, comprehension, choice, induction, etc is needed? Since \(Z_2\) is a natural and sufficient environment for many mathematical theorems, it is an appropriate framework for answering questions raised by the reverse mathematics program. The main (but not the only) distinctions that are made in reverse mathematics concern the amount of the Comprehension Principle that is needed in proving this or that mathematical result. In particular, the role of Arithmetic and \(\Pi^1_1\)-Comprehension Principles is clarified. The basic theory of real numbers can be developed on the basis of Arithmetic Comprehension only, but proofs relying on the notion of a countable ordinal require the \(\Pi^1_1\)-Comprehension Principle. Again, we refer to Simpson (1999 [2009]) for details.