Forcing (mathematics)

In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. It was first used, in 1962, to prove the independence of the continuum hypothesis and the axiom of choice. Forcing was considerably reworked and simplified in the sixties, and is nowadays a basic technique.

Forcing is equivalent to the method of Boolean-valued models, which is conceptually more natural and intuitive, but usually much more difficult to apply.

Intuitively, forcing consists of expanding the set theoretical universe V to a larger universe V*. In this bigger universe, for example, one might have lots of new subsets of ω = {0,1,2,…} that weren't there in the old universe, and thereby violate the continuum hypothesis. While impossible on the face of it, this is just another version of Cantor's "paradoxes" about infinity. In principle, one could consider V*=V×{0,1}, identify x∈V with (x,0), and then introduce an expanded membership relation involving the "new" sets of the form (x,1). Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set, and allowing for fine control over the properties of the expanded universe.

Contents

1 Forcing posets 2 Countable transitive models and generic filters 3 Forcing 4 Consistency 5 Cohen forcing 6 The countable chain condition 7 Easton forcing 8 Random reals 9 Boolean-valued models 10 Meta-mathematical explanation 11 External links 12 References

Forcing posets

A forcing poset is an ordered triple

(P, ≤, 1)

where "≤" is a reflexive, transitive binary relation on P, 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 [3.1415926,3.1415927] provides more information about the number π than [3.1,3.2] 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 noticeably 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}.

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

• Name(0)={};
• Name(α + 1) = {(u, p) : u ∈ Name(α) and p ∈ 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 induction.

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 induction.) 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 = [0,1] 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.

Countable transitive models and generic filters

The key step in forcing is, given a ZFC universe V, to find appropriate G not in V. The resulting class of all interpretations of P-names will turn out to be a model of ZFC, properly extending the original V (since G∉V).

Instead of working with V, one considers a countable transitive model M with (P,≤,1)∈M. By model, we mean a model of set theory, either of all of ZFC, or a model of a large but finite subset of the ZFC axioms, or some variant thereof. Transitivity means that if x∈y∈M, then x∈M. The Mostowski collapsing theorem says this can be assumed if the membership relation is well-founded. The effect of transitivity is that membership and other elementary notions can be handled intuitively. Countability of the model relies on the Löwenheim-Skolem theorem. Note that we don't actually want or need to produce a model M for all of ZFC (or ZFC + ~CH), but merely the finite portion we subsequently use in the proof; Gödel's second incompleteness theorem prevents us creating a model for the whole theory.

Since M is a set, there are sets not in M - this follows from Russell's paradox. The appropriate set G to pick, and adjoin to M, is a generic filter on P. This means that G⊆P is a filter :

• 1∈G ;
• if p≥q∈G, then p∈G ;
• if p,q∈G, then ∃r∈G, r≤p and q ;

and that G is generic, meaning

• if D∈M is a dense subset of P (that is, p∈P implies ∃q∈D, q≤p) then G∩D≠0 .

The existence of such a G is straightforward. Because M is really countable, one can enumerate the dense subsets of P in M as D₀, D₁, …. (In general, M thinks the number of such dense subsets is uncountable, and hence this enumeration is not in M. This is a version of the Skolem paradox.) By assumption, there exists p₀∈D₀. Then by density, there exists p₁≤p₀ with p₁∈D₁. Repeating, one gets …≤p₂≤p₁≤p₀ with p_i∈D_i. Then G={ q∈P : ∃i, q ≥ p_i } is a generic filter.

In fact, slightly more is true: given a condition p ∈ P, one can find a generic filter G such that p ∈ G. This holds because one can start with p₀ ≤ p.

If P has only countably many dense subsets, then the above shows that one can pick G ∈ M. This is the trivial case. Minimal elements in P are also trivial, since if D is dense and p is minimal, then since the only element q ≤ p is p itself, p ∈ D. Thus, any filter containing even one minimal element is generic, and one can again choose G ∈ M.

Forcing

Given a generic filter G⊆P, one proceeds as follows. The subclass of P-names in M is denoted M^(P). Let M[G]={val(u,G):u∈M^(P)}. To reduce the study of the set theory of M[G] 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,…,u_n) (read "p forces φ") where p is a condition, φ is a formula in the forcing language, and the u_i are names, to mean that if G is a generic filter containing p, then M[G] ⊨ φ(val(u₁,G),…,val(u_n,G)). The special case 1 ⊩ φ is often written P ⊩ φ or ⊩ φ. Such statements are true in M[G] 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[G] are really properties of M, and the verification of ZFC in M[G] becomes straightforward. This is usually summarized as three key properties:

• Truth: M[G] ⊨ φ(val(u₁,G),…,val(u_n,G)) iff it is forced by G, that is, for some condition p ∈ G, p ⊩ φ(u₁,…,u_n).
• Definability: The statement "p ⊩ φ(u₁,…,u_n)" is definable in M.
• Coherence: If p ⊩ φ(u₁,…,u_n) and q ≤ p, then q ⊩ φ(u₁,…,u_n).

Consistency

The above can be summarized by saying the fundamental consistency result is that given a forcing poset P, we may assume that there exists a generic filter G, not in the universe V, such that V[G] is again a set theoretic universe, modelling ZFC. Furthermore, all truths in V[G] can be reduced to truths in V regarding the forcing relation.

Both styles, adjoining G to a countable transitive model M or to the whole universe V, are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, and no mention of set or class models is made. This was Cohen's original method, and in one elaboration, it becomes the method of Boolean-valued analysis.

Cohen forcing

The simplest nontrivial forcing poset is ( Fin(ω,2) , ⊇ , 0 ), the finite partial functions from ω to 2={0,1} under reverse inclusion. That is, a condition p is essentially two disjoint finite subsets p⁻¹[1] and p⁻¹[0] of ω, to be thought of as the "yes" and "no" parts of p, with no information provided on values outside the domain of p. q is stronger than p means that q ⊇ p, in other words, the "yes" and "no" parts of q are supersets of the "yes" and "no" parts of p, and in that sense, provide more information.

Let G be a generic filter for this poset. If p and q are both in G, then p∪q is a condition, because G is a filter. This means that g=⋃G is a well-defined partial function from ω to 2, because any two conditions agree on their common domain.

g is in fact a total function. Given n ∈ ω, let D_n={ p : p(n) is defined }, then D_n is dense. (Given any p, if n is not in p’s domain, adjoin a value for n, the result is in D_n.) A condition p ∈ G∩D_n has n in its domain, and since p ⊆ g, g(n) is defined.

Let X=g⁻¹[1], the set of all "yes" members of the generic conditions. It is possible to give a name for X directly. Let X = { ( nˇ , p ) : p(n)=1 }, then val( X , G ) = X. Now suppose A⊆ω in V. We claim that X≠A. Let D_A = { p : ∃n, n∈dom(p) and p(n)=1 iff n∉A }. D_A is dense. (Given any p, if n is not in p’s domain, adjoin a value for n contrary to the status of "n∈A".) Then any p∈G∩D_A witnesses X≠A. To summarize, X is a new subset of ω, necessarily infinite.

Replacing ω with ω×ω₂, that is, consider instead finite partial functions whose inputs are of the form (n,α), with n<ω and α<ω₂, and whose outputs are 0 or 1, one gets ω₂ new subsets of ω. They are all distinct, by a density argument: given α<β<ω₂, let D_α,β={p:∃n, p(n,α)≠p(n,β)}, then each D_α,β is dense, and a generic condition in it proves that the αth new set disagrees somewhere with the βth new set.

This is not yet the falsification of the continuum hypothesis. One must prove that no new maps have been introduced which map ω onto ω₁ or ω₁ onto ω₂. For example, if one considers instead Fin(ω,ω₁), finite partial functions from ω to ω₁, the first uncountable ordinal, one gets in V[G] a bijection from ω to ω₁. In other words, ω_{1 has collapsed, and in the forcing extension, is a countable ordinal.}

The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does not collapse cardinals. It turns out to follow from a combinatorial property this poset has: all of its antichains are countable.

The countable chain condition

An antichain A of P is a subset such that if p and q are in A, then p and q are incompatible (written p ⊥ q), meaning there is no r in P such that r ≤ p and r ≤ q. In the Borel sets example, incompatibility means p∩q has measure zero. In the finite partial functions example, incompatibility means that p∪q is not a function, in other words, p and q assign different values to some domain input.

P has the countable chain condition (c.c.c.) is that assertion that every antichain is countable. (The name, which is obviously inappropriate, is a holdover from older terminology. Attempts to fix this have not succeeded.)

It is easy to see that Bor(I) has the c.c.c., because the measures add up to at most 1. Fin(E,2) is also c.c.c., but the proof is more difficult.

Given an uncountable subfamily W ⊆ Fin(E,2), shrink W to a uncountable subfamily W₀ of sets of size n, for some n<ω. If p(e₁)=b₁ for uncountably many p ∈ W₀, shrink to this uncountable subfamily W₁, and repeat, getting a finite set { (e₁,b₁) , … , (e_k,b_k) }, and an uncountable family W_k of incompatible conditions of size n−k such that every e is in at most countably many dom(p) for p ∈ W. Now pick an arbitrary p ∈ W_k, and pick from W_k any q that is not one of the countably many members that have a domain member in common with p. Then p ∪ { (e₁,b₁) , … , (e_k,b_k) } and q ∪ { (e₁,b₁) , … , (e_k,b_k) } are compatible, so W is not an antichain. In other words, Fin(E,2) antichains are countable.

The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A maximal antichain A is one that cannot be extended and still be an antichain. This means every element of p ∈ P is compatible with some member of A. Their existence follows from Zorn's lemma. Given a maximal antichain A, let D = { p : p≤q, some q∈A }. D is dense, and G∩D≠0 iff G∩A≠0. Conversely, given a dense set D, Zorn's lemma shows there exists a maximal antichain A⊆D, and then G∩D≠0 iff G∩A≠0.

Assume P is c.c.c. Given x,y ∈ V, with f:x→y in V[G], one can approximate f inside V as follows. Let u be a name for f (by the definition of V[G]) and let p be a condition which forces u to be a function from x to y. Define a function F whose domain is x by F(a) = { b : ∃ q ≤ p, q forces u(aˇ) = bˇ }. By definability of forcing, this definition makes sense within V. By coherence of forcing, different b’s come from incompatible p’s. By c.c.c., F(a) is countable.

In summary, f is unknown in V, since it depends on G, but it is not wildly unknown for a c.c.c. forcing. One can identify a countable set of guesses for what the value of f is at any input, independent of G.

This has the following very important consequence. If in V[G], f:α→β is a surjection from one infinite ordinal to another, then there is a corresponding surjection in V. In particular, cardinals cannot collapse. The conclusion is that 2^{ℵ₀ ≥ ℵ₂ in V[G].}

Easton forcing

The exact value of the continuum in the above Cohen model, and variants like Fin(ω × κ , 2) for cardinals κ in general, was worked out by Robert M. Solovay, who also worked out how to violate GCH (the generalized continuum hypothesis), for regular cardinals only, a finite number of times. For example, in the above Cohen model, if CH holds in V, then 2^{ℵ₀ = ℵ₂ holds in V[G].}

W. B. Easton worked out the infinite and proper class version of violating the GCH for regular cardinals, basically showing the known restrictions (monotonicity, Cantor's theorem, and König's theorem) were the only ZFC provable restrictions.

Easton's work was notable in that it involved forcing with a proper class of conditions. In general, the method of forcing with a proper class of conditions will fail to give a model of ZFC. For example, Fin ( ω × On , 2 ), where "On" is the proper class of all ordinals, will make the continuum a proper class. Fin ( ω , On ) will introduce a countable enumeration of the ordinals. In both cases, the resulting V[G] is visibly not a model of ZFC.

At the time, it was thought that more sophisticated forcing would also allow arbitrary variation in the powers of singular cardinals. But this has turned out to be a difficult, subtle and even surprising problem, with several more restrictions provable in ZFC, and with the forcing models depending on the consistency of various large cardinals. Many open problems remain.

Random reals

In the Borel sets ( Bor(I) , ⊆ , I ) example, the generic filter converges to a real number r, called a random real. A name for the decimal expansion of r (in the sense of the canonical set of decimal intervals that converge to r) can be given by letting r = { ( Eˇ , E ) : E = [ k⋅10⁻ⁿ , (k+1)⋅10⁻ⁿ ], 0≤k<10ⁿ }. This is, in some sense, just a subname of G.

To recover G from r, one takes those Borel subsets of I that "contain" r. Since the forcing poset is in V, but r is not in V, this containment is actually impossible. But there is a natural sense in which the interval [.5,.6] in V "contains" a random real whose decimal expansion begins .5. This is formalized by the notion of "Borel code".

Every Borel set can, nonuniquely, be built up, starting from intervals with rational endpoints and applying the operations of complement and countable unions, a countable number of times. The record of such a construction is called a Borel code. Given a Borel set B in V, one recovers a Borel code, and then applies the same construction sequence in V[G], getting a Borel set B*. One can prove that one gets the same set independent of the construction of B, and that basic properties are preserved. For example, if B⊆C, then B*⊆C*. If B has measure zero, then B* has measure zero.

So given r, a random real, one can show that G = { B (in V) : r∈B* (in V[G]) }. Because of the mutual interdefinability between r and G, one generally writes V[r] for V[G].

A different interpretation of reals in V[G] was provided by Dana Scott. Rational numbers in V[G] have names that correspond to countably many distinct rational values assigned to a maximal antichain of Borel sets, in other words, a certain rational-valued function on I = [0,1]. Real numbers in V[G] then correspond to Dedekind cuts of such functions, that is, measurable functions.

Boolean-valued models

Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In it, any statement is assigned a truth value from some infinite Boolean algebra, rather than just a true/false value. Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model which contains this ultrafilter, which can be understood as a model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in appropriate way, we can get a model that has the desired property. In it, only statements which must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property).

Meta-mathematical explanation

In forcing we usually seek to show some sentence is consistent with ZFC (or optionally some extension of ZFC). One way to interpret the argument is that we assume ZFC is consistent and use it to prove ZFC combined with our new sentence is also consistent.

Each "condition" is a finite piece of information - the idea is that only finite pieces are relevant for consistency, since by the compactness theorem a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. Then, we can pick an infinite set of consistent conditions to extend our model. Thus, assuming consistency of set theory, we prove consistency of the theory extended with this infinite set.

External links

• Tim Chow's newsgroup article Forcing for dummies (http://www-math.mit.edu/~tchow/mathstuff/forcingdum) is a good introduction to the concepts of forcing; it covers the main ideas, omitting technical details

References

• K. Kunen, Set Theory: An Introduction to Independence Proofs, North Holland