INTRODUCTION

Properness is a property of forcing posets which generalises the countable chain condition, is preserved under countable support iterations, and which, when applied to iterations of posets of cardinality ℵ1. (the posets one usually wants to iterate) guarantees the preservation of all cardinals. The notion was formulated by Saharon Shelah, and arose from his study of Ronald Jensen's technique of iterated Souslin forcing (see [DeJo]) and from his attempts (eventually successful) to show that the GCH does not resolve the Whitehead Problem (see [Ek], together with §4 of these notes). In order to motivate the notion, let us review some basic forcing theory, which we will in any case need later.

If ℙ is a poset (we always assume that ℙ has a maximum element, 1, and that for each p ∈ ℙ there are q, r ∈ ℙ such that q ≤ p, r ≤ p, and q, r are incomparable in ℙ), there is a uniquely defined complete boolean algebra BA(ℙ) which extends ℙ as a partially ordered set and has ℙ as a dense subset (in the usual, poset sense). We denote by V(ℙ) the boolean universe V(BA(ℙ)) (see [Je] for basic forcing theory).