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So far we have seen a way of building inner models of ZFC from the existing models. Forcing is a way of extending a given model of a portion of ZFC to another one. In fact, forcing is technically speaking, only applicable to countable transitive models (we shall discuss why later). The logic behind this process is as follows.
The last sections of Chapter 17 show why the forcing techniques that we have at N1 cannot possibly work at singular cardinals. Similar concerns apply to the successors of singular cardinals. For example, if we wish to study values of various cardinal invariants at κ, then we had better make sure that they are not trivially equal to κ+. Therefore we wish to work in the context of 2κ > κ+. If, in addition, we have that κ is a strong limit cardinal, then we are automatically dealing with the failure of SCH and so with large cardinals. In this situation, the cardinal invariants at κ+ are also affected. This situation presents many challenges and at this moment there is no unique technique or an axiom that deals with it. However, some techniques have emerged in recent years, two of which will be described below.
In this chapter we see how the technique of forcing may be used to obtain many more independence results, involving important problems such as the Souslin problem. We see our first forcing axiom, Martin’s Axiom.
A new chapter in the theory of forcing was opened by Shelah’s discovery of proper forcing in . The novelty of Shelah’s approach is a move from infinite combinatorics to the technique of elementary submodels, which has since integrated set theory to the extent that it is hard to imagine a forcing argument that does not use it. A major reference on the technique and results about proper forcing is Shelah’s book . Throughout this chapter, we fix an uncountable regular cardinal χ which is so large that all arguments relative to the forcing notions we discuss are contained in the set H(χ) of all sets x with |TC(x)| < χ.
There is a natural order on the cardinals, induced from the ordinals. So, for example, 0 < 1 < 2 < … < ω < ω1. Notice that the order on the cardinals can also be defined by saying that κ < λ if there is an injection from κ to λ but not the other way around.
A very well known early instance that shows a limitation of Martin’s Axiom is Richard Laver’s proof of the consistency of the Borel Conjecture . Laver proved that this problem, which seems along the lines of those that have been solved using Martin’s Axiom, on cardinal invariants of the continuum, cannot be answered using Martin’s Axiom. A subset X of R is said to have strong measure zero if for every sequence (εn)n of positive real numbers, there is a sequence of intervals (In)n such that lg(In) < εn and X ⊆ ∪n<ωIn. It is clear that all countable subsets of R have strong measure 0 and the Borel Conjecture from  states that the only strong measure zero sets are the countable ones.
Set theory has a dual nature as a subject which is on the one hand foundational and on the other, mathematical. Its foundational background comes from the context in which after the Hilbert programme there came a search for axioms that would encapsulate the entire mathematics the way that, say, the axioms of Euclidean geometry in David Hilbert’s rendering of it, encapsulate that subject. The late nineteenth century was the time when many mathematical developments had made it clear that it was no longer possible to do mathematics without being aware of the foundations behind it, as paradoxes were starting to appear. Hence the Hilbert programme specifically set the goal of finding foundations for mathematics. Set theory has soon emerged as an acceptable, if not entirely perfect answer–as we shall explain later on. The subject of set theory is probably still best known for its connections with foundations.
In this short chapter we discuss some forcing notions that although not that much newer than Cohen forcing, already form part of an advanced set theory investigation. Yet, these are so classical that one should certainly know them even before thinking of the advances that came later. In addition to the forcing notions studied in this chapter, another one is Easton forcing, given in §17.1, and yet another the notion of random reals which we let the reader discover from  or .
The iteration theorems we presented so far give a wealth of results about the subsets of ω and, to some extent, ω1. For example, a typical result obtained by the application of iterated proper forcing is the construction of a model in which ω1 = □ < ■ = ω2, where □ and ■ are cardinal invariants of the continuum. One can consult the book  by Tomek Bartoszyński and Haim Judah for various instances of this method. The difficulties of generalising this context to the larger cardinals, even of the form κ+ for κ satisfying κ<κ = κ, turn out to be substantial. In the above mentioned exposition of iterated forcing Baumgartner [5, §4] wrote
To develop more knowledge of forcing we have to go beyond the theory of ZFC and discuss large cardinals.We explore why this is the case and introduce some concrete large cardinal notions, including the supercompact cardinal.We shall, however, motivate that by going through a well known example of a large cardinal forcing first.
We start with a lemma that gives an equivalent definition of a well ordering, and which is often used in practice when we check whether a set is well ordered or not. We use the notation (L, <) of the strict order and (L, ≤) of the order which is not necessarily strict, interchangeably, since each of these objects is induced by the other. All theorems in the theory of order translate between these versions in a straightforward manner.
In this part we discuss results in set theory that have been obtained since the invention of forcing and up to now. So, not everything in this part is exactly new, but it goes beyond the popular perception of what set theory is, which seems to stop at Cohen’s work. That is where this part of the book starts.