In set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe.

Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as computability theory. Descriptive set theory uses the notions of forcing from both computability theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define genericity directly without mention of forcing.

Intuition

of them), identified with subsets of natural numbers, that were not there in the old universe, and thereby violate the continuum hypothesis.

In order to intuitively justify such an expansion, it is best to think of the "old universe" as a model. By the Löwenheim–Skolem theorem can be chosen to be a "bare bones" model that is externally countable. Specifically, there is an ordinal that "plays the role of the cardinal.)

(more generally, that cardinal collapse should be given a (non-unique) name, just like in a simple field extension will correspond to an actual model.

is taken to be an arbitrary (e.g. the countability to describe".

to be a generic set.) of a generic set is "forced" to hold under some forcing condition is indeed a model that satisfies the desired properties.

Cohen's original technique, now called ramified forcing, is slightly different from the unramified forcing expounded here. Forcing is also equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.

The role of the model

must in fact be a standard transitive model. A standard transitive model can be obtained from any standard model through the Mostowski collapse lemma.

(any axiomatization has at least one axiom schema, and thus an infinite number of axioms), the existence of which is guaranteed by the reflection principle. As the goal of a forcing argument is to prove consistency results, this is enough since any inconsistency in a theory must manifest with a derivation of a finite length, and thus involve only a finite number of axioms.

Forcing conditions and forcing posets

Each forcing condition can be regarded as a finite adjoined to the model. There are many different ways of providing information about an object, which give rise to different forcing notions. A general approach to formalizing forcing notions is to regard forcing conditions as abstract objects with a poset structure.

A forcing poset is a preorder are the forcing conditions (or just conditions is stronger provides more information about the number must satisfy the splitting condition:

• .

.

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.

Examples

be any infinite set. In Cohen's original formulation of forcing, each forcing condition is a finite and do not appear in the same condition). This forcing notion is usually called Cohen forcing.

under reverse to get two new forcing conditions, incompatible with each other.

is the collection of Borel subsets having non-zero Lebesgue measure. The generic object associated with this forcing poset is a random real number with measure 1, provided that the Borel subset is "described" in the original unexpanded universe (this can be formalized with the concept of Borel codes). Each forcing condition can be regarded as a random event with probability equal to its measure. Due to the ready intuition this example can provide, probabilistic language is sometimes used with other divergent forcing posets.

Generic filters

is commonly considered to be.

should be a generic filter. The "filter is a set of all true forcing conditions:

" means:

• .

follows from the Rasiowa–Sikorski lemma.

P-names and interpretations

is the class-names of the form

, the interpretation or valuation-names is given by

-names are, in fact, an expansion of the universe-name

.

:

.

Rigorous definitions

may be defined by transfinite recursion the empty set the successor ordinal the power-set a limit ordinal, define the following hierarchy:

-names is defined as

can similarly be defined with a hierarchical construction.

Forcing

. Let

, one works with the "forcing language", which is built up like ordinary first-order logic-names as constants.

is.

What is important is that this external is equivalent to an internal, defined by transfinite induction (specifically \in-induction becomes straightforward. This is usually summarized as the following three key properties:

• Truth if and only if.

• Definability.

• Coherence.

Internal definition

..

can be defined recursively as follows:

-names with lesser ranks, so transfinite induction allows the definition to go through.

) automatically satisfies Definability also satisfies Truth and Coherence as the elementary symbols to be a generic filter.

described at the top of this section:

".

Consistency

involving the forcing relation.

, are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, in which 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

under reverse, to be thought of as the "yes" and "no" parts of , and in that sense, provide more information.

is a well-defined partial function agree on their common domain.

is defined.

directly. Let

. Let

, necessarily infinite.

, let

is dense, and a generic condition in it proves that the αth new set.

, the first uncountable ordinal a bijection 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. For this, a sufficient combinatorial property is that all of the antichains of the forcing poset are countable.

The countable chain condition

A (strong) antichain are incompatible assign different values to some domain input.

is said to satisfy the countable chain condition is countable. (The name, which is obviously inappropriate, is a holdover from older terminology. Some mathematicians write "c.a.c." for "countable antichain condition".)

satisfies the c.c.c., but the proof is more difficult.

-antichains are countable.

The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A maximal. The existence of a maximal antichain follows from Zorn's Lemma, let

.

, by

is countable.

.

.

Easton forcing

in general, was worked out by Robert M. Solovay (the generalized continuum hypothesis), for regular cardinals.

William B. Easton for regular cardinals, basically showing that the known restrictions, (monotonicity, Cantor's theorem and König's theorem-provable restrictions (see Easton's theorem).

.

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

Random reals

(a smaller set in the context of inclusion is a smaller set in the ordering, and represents a condition with more information). There are two types of important dense sets:

1. is the diameter.

2. is dense.

. In this ordering, this means that any filter is closed under finite intersection. Therefore, by Cantor's intersection theorem, the intersection of all.

.

of topological closures.

is a Borel set of measure 1", then by the truth property of forcing

that satisfies

,

.

Every Borel set can be (non-uniquely) built up, starting from intervals with rational endpoints and applying the operations of complement and countable union, a countable number of times. The record of such a construction is called a Borel code is injective.

.

.

, a random real, one can show that

.

was provided by Dana Scott 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 these, any statement is assigned a truth value from some complete atomless 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 that contains this ultrafilter, which can be understood as a new model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in an appropriate way, we can get a model that has the desired property. In it, only statements that 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 that some sentence is consistent combined with the 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 extended by this infinite set.

Logical explanation

By Gödel's second incompleteness theorem. Such problems are known as problems of relative consistency, one of which proves

The general schema of relative consistency proofs follows. As any proof is finite, it uses only a finite number of axioms:

can verify the validity of this proof. This is provable by induction on the length of the proof.

Then resolve

By proving the following

it can be concluded that

which is equivalent to

of course.)

.

.

.

(the axiom of constructibility).

See also

• List of forcing notions

• Nice name

Notes

References

Bibliography