Basis theorem (computability)

In computability theory, there are a number of basis theorems. These theorems show that particular kinds of sets always must have some members that are, in terms of Turing degree, not too complicated. One family of basis theorems concern nonempty effectively closed sets (that is, nonempty $\Pi _{1}^{0}$ sets in the arithmetical hierarchy); these theorems are studied as part of classical computability theory. Another family of basis theorems concern nonempty lightface analytic sets (that is, $\Sigma _{1}^{1}$ in the analytical hierarchy); these theorems are studied as part of hyperarithmetical theory.

Effectively closed sets

Effectively closed sets are a topic of study in classical computability theory. An effectively closed set is the set of all paths through some computable subtree of the binary tree $2^{<\omega }$. These sets are closed, in the topological sense, as subsets of the Cantor space $2^{\omega }$, and the complement of an effective closed set is an effective open set in the sense of effective_Polish_spaces. Stephen Cole Kleene proved in 1952 that there is a nonempty, effectively closed set with no computable point (Cooper 1999, p. 134). There are basis theorems that show there must be points that are not "too far" from being computable, in an informal sense.

These basis theorems include (Cooper 1999, p. 134):

• The Low basis theorem: each nonempty $\Pi _{1}^{0}$ set has a member that is of r.e. degree.
• The hyperimmune-free basis theorem: each nonempty $\Pi _{1}^{0}$ set has a member that is of hyperimmune-free degree.
• The r.e. basis theorem: each nonempty $\Pi _{1}^{0}$ set has a member that is of r.e. degree.

Lightface analytic sets

There are also basis theorems for lightface $\Sigma _{1}^{1}$ sets. These basis theorems are studied as part of hyperarithmetical theory. One theorem is the Gandy basis theorem, a variation of the low basis theorem. The Gandy basis theorem shows that each nonempty $\Sigma _{1}^{1}$. set has an element that is hyperarithmetically low, that is, which has the same hyperdegree as [[Kleene's O|${\mathcal {O}}$.

References

• Cooper, S. B. (1999). "Local degree theory", in Handbook of Computability Theory, pp. 121–153

External links

• Simpson, S. "A survey of basis theorems", slides from Computability Theory and Foundations of Mathematics, Tokyo Institute of Technology, February 18–20, 2013.