Turing Degrees, Dovetailing, Godel

In the last lecture we defined Turing degrees, which are basically sets of languages. A Turing degree consists of a set of languages that can all be reduced to another language in the same set using a Turing reduction.

The set of computable languages is called degree 0. The set of languages equivalent to the halting problem is just above degree 0. A natural question to now ask is are there degrees between these two?

It turns out the answer is yes. There are infinitely many Turing degrees between degree 0 and the halting degree. What it means to be “between” these two degrees is that if we have a language L in this intermediate degree, the following relation holds -

\[L \le {}_{T} HALT\] \[L \nle {}_{T} \text{degree} 0\] \[HALT \nle {}_{T} L\]

This means that the language L is reducible to the halting problem. That is, you can build a Turing machine to recognize L given an oracle for the halting problem. However, the language is not reducible to any language in degree 0, and the halting problem is not reducible to language L.