This page is supposed to contain some pointers to references in Axiomatic Set Theory.
A very useful source of references is provided by the 3 volumes of Gödel's Collected Works, and the introduction, written by Scott, of the book on Boolean Valued set models, by Bell.
The basic intuition being constructible sets comes from the ramified hierarchy, introduced by Russell: one introduces first only subsets defined by quantification over natural numbers, they are subsets of the first level. The subsets of the second level are obtained by allowing quantification over subsets of the first level, subsets of the third level by allowing quantification over subsets of the second level, etc... The idea of Godel was to extend this transfinitely: a subset of level omega is by definition a subset of some finite levels, etc... The essence of Godel's consistency proof is that this process stops at level omega_1. The first exercice in ExIII gives a direct argument why the ramified hierarchy has to stop at the level omega_1.
See also the following discussion on the Continuum Hypothesis CH and on the Axiom of Choice AC
I found it remarkable that most models showing the independence of AC are based, more or less implicitely, on the following examples of Russell: given an infinite number of boots one can define a choice function (for instance taking always the left boot) but this is not possible with an infinite number of socks. The first such model, by Fraenkel, introduces an infinite number of atoms pn, qn, with pn indiscernible from qn.