Forcing (computability)

Forcing in computability theory is a modification of Paul Cohen's original set-theoretic technique of forcing to deal with computability concerns.

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. Both techniques are described as a relation (customarily denoted ${\displaystyle \Vdash }$) 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, computability-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 computability. But while the machinery may be somewhat different, computability-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 ${\displaystyle 2^{\omega }}$. In other words, a function that maps each integer to either 0 or 1.

string

an element of ${\displaystyle 2^{<\omega }}$. In other words, a finite approximation to a real.

notion of forcing

A notion of forcing is a set ${\displaystyle P}$ and a partial order on ${\displaystyle P}$, ${\displaystyle \succ _{P}}$ with a greatest element${\displaystyle 0_{P}}$.

condition

An element in a notion of forcing. We say a condition ${\displaystyle p}$ is stronger than a condition ${\displaystyle q}$ just when ${\displaystyle q\succ _{P}p}$.

compatible conditions

Given conditions ${\displaystyle p,q}$ say that ${\displaystyle p}$ and ${\displaystyle q}$ are compatible if there is a condition ${\displaystyle r}$ such that with respect to ${\displaystyle r}$, both ${\displaystyle p}$ and ${\displaystyle q}$ can be simultaneously satisfied if they are true or allowed to coexist.

${\displaystyle p\mid q}$

means ${\displaystyle p}$ and ${\displaystyle q}$ are incompatible.

Filter

A subset ${\displaystyle F}$ of a notion of forcing ${\displaystyle P}$ is a filter if ${\displaystyle p,q\in F\implies p\nmid q}$, and ${\displaystyle p\in F\land q\succ _{P}p\implies q\in F}$. In other words, a filter is a compatible set of conditions closed under weakening of conditions.

Ultrafilter

A maximal filter, i.e., ${\displaystyle F}$ is an ultrafilter if ${\displaystyle F}$ is a filter and there is no filter ${\displaystyle F'}$ properly containing ${\displaystyle F}$.

Cohen forcing

The notion of forcing ${\displaystyle C}$ where conditions are elements of ${\displaystyle 2^{<\omega }}$ and ${\displaystyle (\tau \succ _{C}\sigma \iff \sigma \supset \tau }$)

Note that for Cohen forcing ${\displaystyle \succ _{C}}$ is the reverse of the containment relation. This leads to an unfortunate notational confusion where some computability theorists reverse the direction of the forcing partial order (exchanging ${\displaystyle \succ _{P}}$ with ${\displaystyle \prec _{P}}$, 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 ${\displaystyle \sigma }$ is stronger than ${\displaystyle \tau }$ when ${\displaystyle \sigma }$ agrees with everything ${\displaystyle \tau }$ 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 ${\displaystyle \tau \succ _{C}\sigma }$ then ${\displaystyle \sigma }$ tells us the value of the real at more places.

In a moment we will define a relation ${\displaystyle \sigma \Vdash _{P}\psi }$ (read ${\displaystyle \sigma }$ forces ${\displaystyle \psi }$) that holds between conditions (elements of ${\displaystyle P}$) and sentences, but first we need to explain the language that ${\displaystyle \psi }$ is a sentence for. However, forcing is a technique, not a definition, and the language for ${\displaystyle \psi }$ will depend on the application one has in mind and the choice of ${\displaystyle P}$.

The idea is that our language should express facts about the object we wish to build with our forcing construction.

Forcing relation

The forcing relation ${\displaystyle \Vdash }$ was developed by Paul Cohen, who introduced forcing as a technique for proving the independence of certain statements from the standard axioms of set theory, particularly the continuum hypothesis (CH).

The notation ${\displaystyle V\Vdash \phi }$ is used to express that a particular condition or generic set forces a certain proposition or formula ${\displaystyle \phi }$ to be true in the resulting forcing extension. Here's ${\displaystyle V}$ represents the original universe of sets (the ground model), ${\displaystyle \Vdash }$ denotes the forcing relation, and ${\displaystyle \phi }$ is a statement in set theory. When ${\displaystyle V\Vdash \phi }$, it means that in a suitable forcing extension, the statement ${\displaystyle \phi }$ will be true.

References

• Fitting, Melvin (1981). Fundamentals of generalized recursion theory. Studies in Logic and the Foundations of Mathematics. Vol. 105. Amsterdam, New York, and Oxford: North-Holland Publishing Company. pp. 1078–1079. doi:10.2307/2273928. JSTOR   2273928. S2CID   118376273.
• Odifreddi, Piergiorgio (1999). Classical recursion theory. Vol. II. Studies in Logic and the Foundations of Mathematics. Vol. 143. Amsterdam: North-Holland Publishing Company. ISBN   978-0-444-50205-6. MR   1718169.