Reducing Ordinal Recursion

Ordinal Recursion Theory

Ordinal machines and admissible recursion theory

We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: α-computability theory. We compare the new theory to α-recursion theory, which was developed by G. Sacks and his school. For α an admissible ordinal, the basic notions...

An ordinal analysis of admissible set theory using recursion on ordinal notations ∗ Jeremy Avigad

The notion of a function from N to N defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well-orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke-Platek admissible se...

An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations

The notion of a function from N to N defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well-orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke-Platek admissible se...

Reducing Recursion to Iteration by Algebraic Extension