0 - An introduction

Summary

In this book we prove some results about Zermelo–Fraenkel (ZF) set theory. Before discussing the main result (Theorem 0.1) below we introduce some concepts, and notation, together with just a few remarks to try and put the results in an historical setting as well as illuminating, we hope, their contemporary context for someone who is not a set–theorist.

The Constructible Universe of sets, L, was invénted by Kurt Gödel in 1938 in order to prove the consistency of the Axiom of Choice (AC) and the Generalised Continuum Hypothesis (that 2^κ = κ⁺ for all infinite cardinals, the “GCH”) with the other axioms of set theory, i.e. ZF. L, the smallest model of ZF containing all the ordinals, is constructed by starting with the empty set, ø, and putting in L only what must be there by virtue of the axioms of ZF. At each stage of the construction, any newly constructed element can be defined by a first–order formula using previously defined elements, thus giving L its well–defined structure. Indeed this method implicitly contains the well–ordering of the universe that provides for the truth of AC (and ultimately the GCH in L.

It is possible to inspect the internal structure of this universe more closely, as has recently been done, and the fine structure that arises enables one to have a better ‘grip’ on L than on a general model of ZF or ZF + GCH ; this has led to several results of interest about L.