In the mathematical discipline of 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.
can be chosen to be a "bare bones" model that is externally countable, which guarantees that there will be many subsets (in
(more generally, that cardinal collapse does not occur), and allow fine control over the properties of
[1] [2] Forcing avoids such problems by requiring the newly introduced set
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.
There are many different ways of providing information about an object, which give rise to different forcing notions.
The generic object associated with this forcing poset is a random real number
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.
Under this convention, the concept of "generic object" can be described in a general way.
, one works with the "forcing language", which is built up like ordinary first-order logic, with membership as the binary relation and all the
This is usually summarized as the following three key properties: There are many different but equivalent ways to define the forcing relation
[3] Formally, an internal definition of the forcing relation (such as the one presented above) is actually a transformation of an arbitrary formula
The discussion above can be summarized by the fundamental consistency result that, given a forcing poset
Less commonly seen is the approach using the "internal" definition of forcing, in which no mention of set or class models is made.
is dense, and a generic condition in it proves that the αth new set disagrees somewhere with the
For this, a sufficient combinatorial property is that all of the antichains of the forcing poset are countable.
(the generalized continuum hypothesis), for regular cardinals only, a finite number of times.
William B. Easton worked out the proper class version of violating the
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 fails to give a model of
At one time, it was thought that more sophisticated forcing would also allow an arbitrary variation in the powers of singular cardinals.
However, this has turned out to be a difficult, subtle and even surprising problem, with several more restrictions provable in
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
Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory.
Then we can pick an infinite set of consistent conditions to extend our model.
By Gödel's second incompleteness theorem, one cannot prove the consistency of any sufficiently strong formal theory, such as
In the case of Boolean-valued forcing, the procedure is similar: proving that the Boolean value of