Π01 class

In computability theory, a Π⁰₁ class is a subset of 2^ω of a certain form. These classes are of interest as a technical tool within recursion theory and effective descriptive set theory. They are also used in the application of recursion theory to other branches of mathematics (Cenzer 1999, p. 39).

Definition

The set 2^{< ω} consists of all finite sequences of 0s and 1s, while the set 2^ω consists of all infinite sequences of 0s and 1s (that is, functions from ω = {0, 1, 2, ...} to the set {0,1}).

A tree on 2^{< ω} is a subset of 2^<ω that is closed under taking initial segments. An element f of 2^ω is a path through a tree T on 2^{< ω} if every finite initial segment of f is in T.

A (lightface) Π⁰₁ class is a subset C of 2^ω for which there is a computable tree T such that C consists of exactly the paths through T. A boldface Π⁰₁ class is a subset D of 2^ω for which there is an oracle f in 2^ω and a subtree tree T of 2^{< ω} from computable from f such that D is the set of paths through T.

As effectively closed sets

The boldface Π⁰₁ classes are exactly the same as the closed sets of 2^ω and thus the same as the boldface Π⁰₁ subsets of 2^ω in the Borel hierarchy.

Lightface Π⁰₁ classes in 2^ω (that is, Π⁰₁ classes whose tree is computable with no oracle) correspond to effectively closed sets. A subset B of 2^ω is effectively closed if there is a recursively enumerable sequence ⟨σ_i : i ∈ ω⟩ of elements of 2^{< ω} such that each g ∈ 2^ω is in B if and only if σ_i is an initial segment of B.

Relationship with effective theories

For each effectively axiomatized theory T of first-order logic, the set of all completions of T is a $\Pi _{1}^{0}$ class. Moreover, for each $\Pi _{1}^{0}$ subset S of $2^{\omega }$ there is an effectively axiomatized theory T such that each element of S computes a completion of T, and each completion of T computes an element of S (Jockusch and Soare 1972b).

See also

• Arithmetical hierarchy

References

• Cenzer, Douglas (1999), "$\Pi _{1}^{0}$ classes in computability theory", Handbook of computability theory, Stud. Logic Found. Math., vol. 140, Amsterdam: North-Holland, pp. 37–85, MR 1720779
• Jockush, Carl G.; Soare, Robert I. (1972a), "Degrees of members of $\Pi _{1}^{0}$ classes.", Pacific Journal of Mathematics, 40 (3): 605–616, doi:10.2140/pjm.1972.40.605
• Jockush, Carl G.; Soare, Robert I. (1972b), "$\Pi _{1}^{0}$ Classes and Degrees of Theories", Transactions of the American Mathematical Society, 173: 33–56, doi:10.1090/s0002-9947-1972-0316227-0