Forcing
Given a generic filter G⊆P, one proceeds as follows. The subclass of P-names in M is denoted M(P). Let M={val(u,G):u∈M(P)}. To reduce the study of the set theory of M to that of M, one works with the forcing language, which is built up like ordinary first-order logic, with membership as binary relation and all the names as constants.
Define p φ(u1,…,un) (read "p forces φ in model M with poset P") where p is a condition, φ is a formula in the forcing language, and the ui are names, to mean that if G is a generic filter containing p, then M ⊨ φ(val(u1,G),…,val(un,G)). The special case 1 φ is often written P φ or φ. Such statements are true in M no matter what G is.
What is important is that this "external" definition of the forcing relation p φ is equivalent to an "internal" definition, defined by transfinite induction over the names on instances of u ∈ v and u = v, and then by ordinary induction over the complexity of formulas. This has the effect that all the properties of M are really properties of M, and the verification of ZFC in M becomes straightforward. This is usually summarized as three key properties:
- Truth: M ⊨ φ(val(u1,G),…,val(un,G)) if and only if it is forced by G, that is, for some condition p ∈ G, p φ(u1,…,un).
- Definability: The statement "p φ(u1,…,un)" is definable in M.
- Coherence: If p φ(u1,…,un) and q ≤ p, then q φ(u1,…,un).
We define the forcing relation in V by induction on complexity, in which we simultaneously define forcing of atomic formulas by ∈-induction.
1. p a ∈ b if for any q ≤ p there is r ≤ q such that there is (s, c) ∈ b such that r ≤ s and r a = c.
2. p a = b if p a ⊆ b and p b ⊆ a
- where
- p a ⊆ b if for all q ≤ p and for all (r,c) ∈ a if q ≤ r then q c ∈ b.
3. p ¬ f if there is no q ≤ p such that q f.
4. p f ∧ g if p f and p g.
5. p ∀ x f if p f(a) for any name a where f(a) is result of replacing all free occurrences of x in f by a.
In 1–5 p is an arbitrary condition. In 1 and 2 a and b are arbitrary names and in 3–5 f and g are arbitrary formulas. This definition provides the possibility of working in V without any countable transitive model M. The following statement gives announced definability:
p f if and only if M ⊨ p f.
(Where no confusion is possible we simply write .)
Read more about this topic: Forcing (mathematics)
Famous quotes containing the word forcing:
“no arranged terror: no forcing of image, plan,
or thought:
no propaganda, no humbling of reality to precept:
terror pervades but is not arranged, all possibilities
of escape open: no route shut,”
—Archie Randolph Ammons (b. 1926)
“If we consider what happens in conversation, in reveries, in remorse, in times of passion, in surprises, in the instructions of dreams, wherein often we see ourselves in masquerade,the droll disguises only magnifying and enhancing a real element, and forcing it on our distinct notice,we shall catch many hints that will broaden and lighten into knowledge of the secret of nature.”
—Ralph Waldo Emerson (18031882)
“An empty head is not really empty; it is stuffed with rubbish. Hence the difficulty of forcing anything into an empty head.”
—Eric Hoffer (19021983)