The discovery of forcing. (Q1414982)

This is the text of a talk given by the famous author, a master in the subject, on a conference where his talk was to indicate the background leading up to his work in set theory and to explain in a broad outline the general method of forcing, which he introduced in order to establish various independence results. He admits that the evolution of forcing methods has been so rapid and extensive that he is no longer competent to give any broad survey. However, he remarks that it is gratifying to learn that it has left its mark even on abelian groups, in particular on the Whitehead conjecture. In the introductory section of the paper, the evolution of the Zermelo-Fraenkel system and the contributions of Zermelo, Frege, Skolem, Löwenheim and Fraenkel in this process is discussed; moreover the controversy on Gödel's completeness theorem, which appeared ten years later after Skolem really proved it (in an ``obscure'' Norwegian journal). Cantor's struggle with the continuum hypothesis is described; König's work in this direction is indicated. The second section is a detailed discussion on ``the work of CH and AC'', while the independence of CH is discussed in the third section. A brief description is given of how in 1962 the author began to think about proving independence and how he arrived at it. In this pursuit he arrived at the method of forcing. An elementary statement (or forcing condition) is a finite number of statements of the form \(n\) in \(a\) or \(n\) not in \(a\), where \(n\) is an integer, which are not contradictory. The notion of forcing is that an elementary condition \(P\) forces a statement \(S\) (analogous to implication). We deal with statements \(S\) which have a rank; that means that all variables and all constants occurring in the statement, deal with sets whose rank is bounded by some ordinal. The final definition of forcing is given as: \(P\) forces \(\exists x A(x)\) if, for some \(x\) with the required rank, \(P\) forces \(A(x)\). \(P\) forces \(\forall x A(x)\) if no \(Q>P\) is such that \(Q\) forces the negation, i.e., for some \(y\), \(Q\) forces not \(A(y)\). Here \(P<Q\) means that all the conditions of \(P\) are contained in \(Q\). This article is very informative and useful for those who work in the foundations of mathematics.