The Notion of Forcing

Abstract

In this chapter we present a general technique, called forcing, for extending models of ZFC. The main ingredients to construct such an extension are a model V of ZFC (e.g., $\mathbf{V} =\boldsymbol{\mathop{ \mathrm{L}}\nolimits }$), a partially ordered set $\mathbb{P} = (P,\leq )$ contained in V, as well as a special subset G of P which will not belong to V. The extended model V[G] will then consist of all sets which can be “described” or “named” in V, where the “naming” depends on the set G. The main task will be to prove that V[G] is a model of ZFC as well as to decide (within V) whether a given statement is true or false in a certain extension V[G].

To get an idea of how this is done, think for a moment that there are people living in V. Notice that for these people, V is the unique set-theoretic universe which contains all sets. Now, the key point is that for any statement, these people are in fact able to compute whether the statement is true or false in a particular extension V[G], even though they have almost no information about the set G (in fact, they would actually deny the existence of such a set).