Post's theorem

In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees. As notation we say that a subset <math>X<math> of <math>\omega<math> is <math>\Sigma_n<math> if there is a <math>\Sigma_n<math> formula with free variable <math>n<math> which is true if and only if <math>n<math> is in <math>X<math>.

Formally Post's theorem states:

  • For every <math>n \geq 0<math>, <math>B \in \Sigma_{n+1}<math> if and only if <math>B<math> is a recursively enumerable set with an oracle of some <math>\Pi_n<math> set or, equivalently, some <math>\Sigma_n<math> set.
  • <math>\emptyset^{(n)}<math>, i.e. the n-th Turing jump of the empty set is <math>\Sigma_n<math> complete for every <math>n > 0<math>.

The first result says that the <math>\Sigma_n<math> sets represent sets which are computably enumerable with an oracle in a one lower set. The second result says that the Turing jumps form complete sets of the <math>\Sigma_n<math> (<math>X<math> complete for <math>\Sigma_n<math> means that every other set in <math>\Sigma_n<math> is Turing reducible from <math>X<math>).

As immediate corollaries we get:

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