The method of forcing was invented by Cohen (1963) towards the construction of non-standard models of ZFC, so that “new axioms” could be proved consistent with the standard ones. Our retelling of the basics of forcing found in this chapter is indebted primarily to the user-friendly account found in Shoenfield (1971). The influence of the expositions in Burgess (1978), Jech (1978b), and Kunen (1980) should also be evident.
In outline, the method goes like this: Suppose we want to show that ZFC (sometimes ZF or an even weaker subtheory) is consistent with some weird new axiom, “NA”. Working in the metatheory, one starts with a CTM, M, for ZFC. This is the ground model. One then judiciously chooses a PO set, 〈P, <, 1〉, in M – where we find it convenient to restrict attention to PO sets that have a maximum element (let us call the latter “1”) – and, using the PO set, one constructs a so-called generic set G. Circumstances normally have G obey G ∉ M. The “judicious” aspect of the choice of the PO set will entail that the generic extension, M[G], of the CTM M not only contains G as an element but is a CTM itself that satisfies NA as well (i.e., ⊨M[G] ZFC+NA). Thus, one has a proof in the metatheory that if ZFC is consistent (i.e., if a CTM for ZFC exists), then so is ZFC + NA.
We have said above that “〈P,<, 1〉 ∈ M”. By absoluteness of pair (see Section VI.8), the quoted statement is equivalent to “P ∈ M and<∈ M and 1 ∈ M”.