Combinatorial Set Theory pp 273-293 | Cite as

# 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.*, **V**=**L**), a partially ordered set ℙ=(*P*,≤) 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).

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