Forcing (mathematics) - Forcing Posets

Forcing Posets

A forcing poset is an ordered triple

(P, ≤, 1)

where "≤" is a preorder on P that satisfies following splitting condition

For all p ∈ P, there are q, r ∈ P such that q, r ≤ p with no s ∈ P such that s ≤ q, r

and 1 is a largest element, that is,

p ≤ 1 for all p ∈ P,.

Members of P are called conditions. One reads

p ≤ q

as

p is stronger than q.

Intuitively, the "smaller" condition provides "more" information, just as the smaller interval provides more information about the number π than the interval does.

(There are various conventions here. Some authors require "≤" to also be antisymmetric, so that the relation is a partial order. Some use the term partial order anyway, conflicting with standard terminology, while some use the term preorder. The largest element can be dispensed with. The reverse ordering is also used, most notably by Saharon Shelah and his co-authors.)

Associated with a forcing poset P are the P-names. P-names are sets of the form

{(u,p):u is a P-name and p ∈ P and (some criterion involving u and p)}.

This definition is circular; which in set theory means it is really a definition by transfinite recursion. In long form, one defines

• Name(0) = {};
• Name(α + 1) = a well-defined subset of the power set of (Name(α) × P);
• Name(λ) = ∪{Name(α) : α < λ}, for λ a limit ordinal,

and then defines the class of P-names to be

V(P) = ∪{Name(α) : α is an ordinal}.

The P-names are, in fact, an expansion of the universe. Given x in V, one defines

xˇ

to be the P-name

{(yˇ,1) : y ∈ x}.

Again, this is really a definition by transfinite recursion.

Given any subset G of P, one next defines the interpretation or valuation map from names by

val(u, G) = {val(v, G) : ∃ p ∈ G, (v, p) ∈ u}.

(Again a definition by transfinite recursion.) Note that if 1 is in G, then

val(xˇ, G) = x.

One defines

G = {(pˇ, p) : p ∈ G},

then

val(G,G) = G.

A good example of a forcing poset is

(Bor(I), ⊆, I ),

where I = and Bor(I) are the Borel subsets of I having non-zero Lebesgue measure. In this case, one can talk about the conditions as being probabilities, and a Bor(I)-name assigns membership in a probabilistic sense. Because of the ready intuition this example can provide, probabilistic language is sometimes used with other forcing posets.