### Axiomatic Set Theory

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.

Last modified: Fri Nov 17 16:55:56 MET 2000