Is it possible to prove Turing irreducibility
Let $A_{TM}$ be the language of the Halting problem. Then $\overline{A_{TM}} \nleq_T A_{TM}$, otherwise $A_{TM }$ would be recursive. 

Another well-known theorem by Friedberg–Muchnik proves (using the finite injury priority argument) that there are recursively enumerable languages $A$ and $B$ such that $A \nleq_T B$ and $B \nleq_T A$ (pairwise incomparable). 

R. Soare's book, for example, has more on this subject.