You Can Enter Cantor’s Paradise!

I will try to use a spiralic presentation returning to the same points on higher levels hence repeating ourselves, so that a reader lost somewhere, will not go away empty handed. Also I will assume essentially no particular knowledge and I will say little on the history to which many great mathematicians contributed. 1. Hilbert's first problem Recall (Cantor): • We say that two sets A, B are equinumerous (or equivalent) if there is a one-to-one and onto mapping from A onto B; • The Continuum Hypothesis, CH, is the following statement: every infinite set of reals is either equinumerous with the set Q of rational numbers, or is equinumerous with the set R of all reals; • For a set X, let P(X) denote its power set, i.e., the set of all subsets of X. The Generalized Continuum Hypothesis, GCH, is the statement asserting that for every infinite set X, every subset Y of the power set P(X) is either equinumerous with a subset of X, or is equinumerous with P(X) itself. I think this problem is better understood in the context of: 1.1. Cardinal Arithmetic. Recall (Cantor), that we call two sets A, B equivalent (or equinumerous) if there is a one-to-one mapping from A onto B; the number of elements of A is the equivalence class of A denoted by |A|, we call it also the power or the cardinality of A. Having defined infinite numbers, we can naturally ask ourselves what is the natural meaning of the arithmetical operations and the order. There can be little doubt concerning the order: • |A| ≤ |B| iff A is equivalent to some subset of B. We know that • any two infinite cardinals are comparable so it is really a linear order • any cardinal λ has a successor λ + , which means that • λ < µ ⇔ λ + ≤ µ.