Question

Prove that for any oracle α : N → N, there exists a set A ⊆ N which is Turing equivalent to α, in the sense that A is α-recursive and α is A-recursive. (Thus, we don’t "lose anything" by just considering Turing degrees of sets.)

Answer 1

To prove that for any oracle α : N → N, there exists a set A ⊆ N which is Turing equivalent to α, we need to show that A is α-recursive and α is A-recursive. In other words, A can compute α and α can compute A.

Step-by-step explanation:

To prove that for any oracle α : N → N, there exists a set A ⊆ N which is Turing equivalent to α, we need to show that A is α-recursive and α is A-recursive. In other words, A can compute α and α can compute A.

To do this, we can construct a Turing machine that simulates the behavior of α using the oracle α. This machine takes an input n and uses the oracle to compute α(n). It then outputs the result.

Similarly, we can construct a Turing machine that simulates the behavior of A using the oracle A. This machine takes an input n and uses the oracle to compute A(n). It then outputs the result.