Forcing (recursion theory)
Encyclopedia
Forcing in recursion theory
is a modification of Paul Cohen's
original set theoretic
technique of forcing to deal with the effective concerns in recursion theory
. Conceptually the two techniques are quite similar, in both one attempts to build generic objects (intuitively objects that are somehow 'typical') by meeting dense sets. Also both techniques are elegantly described as a relation (customarily denoted
) between 'conditions' and sentences. However, where set theoretic forcing is usually interested in creating objects that meet every dense set of conditions in the ground model, recursion theoretic forcing only aims to meet dense sets that are arithmetically or hyperarithmetically definable. Therefore some of the more difficult machinery used in set theoretic forcing can be eliminated or substantially simplified when defining forcing in recursion theory. But while the machinery may be somewhat different recursion theoretic and set theoretic forcing are properly regarded as an application of the same technique to different classes of formulas.
real: an element of
. In other words a function that maps each integer to either 0 or 1.
string: an element of
. In other words a finite approximation to a real.
notion of forcing : A notion of forcing is a set
and a partial order on 
, 
with a greatest element 
.
condition: An element in a notion of forcing. We say a condition
is stronger than a condition 
just when 
.
(
incompatible with 
): Given conditions 
say that 
and 
are incompatible (denoted 
) if there is no condition 
with 
and 
. 
is compatible with 
if they are not incompatible.
Filter : A subset
of a notion of forcing 
is a filter if 
and 
. In other words a filter is a compatible set of conditions closed under weakening of conditions.
Ultrafilter : A maximal filter, i.e.,
is an ultrafilter if 
is a filter and there is no filter 
properly containing 
Cohen forcing: The notion of forcing
where conditions are elements of 
and 
)
Note that for Cohen forcing
is the reverse of the containment relation. This leads to an unfortunate notational confusion where some recursion theorists reverse the direction of the forcing partial order (exchanging 
with 
which is more natural for Cohen forcing but is at odds with the notation used in set theory.
is stronger than 
when 
agrees with everything 
says about the object we are building and adds some information of its own. For instance in Cohen forcing the conditions can be viewed as finite approximations to a real and if 
then 
tells us the value of the real on more places.
In a moment we will define a relation
(read 
forces 
) that holds between conditions (elements of 
) and sentences but first we need to explain the language (mathematics) that 
is a sentence for. However, forcing is a technique not a definition and the language for 
will depend on the application one has in mind and the choice of 
.
The idea is that our language should express facts about the object we wish to build with our forcing construction.
Recursion theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
is a modification of Paul Cohen's
Paul Cohen (mathematician)
Paul Joseph Cohen was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory.-Early years:Cohen was born in Long Branch, New Jersey, into a...
original set theoretic
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
technique of forcing to deal with the effective concerns in recursion theory
Recursion theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
. Conceptually the two techniques are quite similar, in both one attempts to build generic objects (intuitively objects that are somehow 'typical') by meeting dense sets. Also both techniques are elegantly described as a relation (customarily denoted
) between 'conditions' and sentences. However, where set theoretic forcing is usually interested in creating objects that meet every dense set of conditions in the ground model, recursion theoretic forcing only aims to meet dense sets that are arithmetically or hyperarithmetically definable. Therefore some of the more difficult machinery used in set theoretic forcing can be eliminated or substantially simplified when defining forcing in recursion theory. But while the machinery may be somewhat different recursion theoretic and set theoretic forcing are properly regarded as an application of the same technique to different classes of formulas.Terminology
In this article we use the following terminology.real: an element of
. In other words a function that maps each integer to either 0 or 1.string: an element of
. In other words a finite approximation to a real.notion of forcing : A notion of forcing is a set
and a partial order on , with a greatest element .condition: An element in a notion of forcing. We say a condition
is stronger than a condition just when . ( incompatible with ): Given conditions say that and are incompatible (denoted ) if there is no condition with and . is compatible with if they are not incompatible. Filter : A subset
of a notion of forcing is a filter if and . In other words a filter is a compatible set of conditions closed under weakening of conditions.Ultrafilter : A maximal filter, i.e.,
is an ultrafilter if is a filter and there is no filter properly containing Cohen forcing: The notion of forcing
where conditions are elements of and )Note that for Cohen forcing
is the reverse of the containment relation. This leads to an unfortunate notational confusion where some recursion theorists reverse the direction of the forcing partial order (exchanging with which is more natural for Cohen forcing but is at odds with the notation used in set theory.Generic objects
The intuition behind forcing is that our conditions are finite approximations to some object we wish to build and that is stronger than when agrees with everything says about the object we are building and adds some information of its own. For instance in Cohen forcing the conditions can be viewed as finite approximations to a real and if then tells us the value of the real on more places.In a moment we will define a relation
(read forces ) that holds between conditions (elements of ) and sentences but first we need to explain the language (mathematics) that is a sentence for. However, forcing is a technique not a definition and the language for will depend on the application one has in mind and the choice of .The idea is that our language should express facts about the object we wish to build with our forcing construction.