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Wikipedia / Alpha recursion theory

In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals. An admissible set denotes a rank of Godel's constructible hierarchy is a model of Kripke–Platek set theory is considered to be fixed.

Definitions

. These sets are said to have some properties:

  • in the definition.

  • A is -recursive (its relative complement.

  • are called -finite and play a similar role to the finite numbers in classical recursion theory.

  • are called -arithmetic.

:

  • A partial function is -recursively-enumerable, or -partial recursive, if and only if.

  • is -recursive. Like in the case of classical recursion theory, any total-recursive.

  • is -arithmetical.

Additional connections between recursion theory and α recursion theory can be drawn, although explicit definitions may not have yet been written to formalize them:

where H, J, K are all α-finite.

A is said to be α-recursive in B reduction procedures such that:

If A is recursive in B. By this definition A (the empty set) if and only if A.

We say A or in other words if every initial portion of A is α-finite.

Work in α recursion

Shore's splitting theorem

Shore's density theorem: Let A, C then there exists a regular α-recursively enumerable set B.

Barwise is from the Lévy hierarchy.

There is a generalization of limit computability functions.

, that at limit computation steps take the limit inferior-Turing machine-Turing machine.

-enumeration degrees.

Relationship to analysis

-recursion theory can be translated into similar results about second-order arithmetic for the language of second-order arithmetic that consists of sets of integers.

In fact, when dealing with first-order logic, the arithmetical and Lévy hierarchies can become interchangeable. For example, a set of natural numbers is a level of the Lévy hierarchy. formula.

References

  • Gerald Sacks, Higher recursion theory, Springer Verlag, 1990 https://projecteuclid.org/euclid.pl/1235422631

  • Robert Soare, Recursively Enumerable Sets and Degrees, Springer Verlag, 1987 https://projecteuclid.org/euclid.bams/1183541465

  • Keith J. Devlin, (p.38), North-Holland Publishing, 1974

  • J. Barwise, Admissible Sets and Structures. 1975

Inline references