We describe Wadge degrees of $\omega$-languages recognizable by deterministic Turing machines. In particular, it is shown that the ordinal corresponding to these degrees is ${\xi }^{\omega }$ where $\xi ={\omega }_1^{\mathrm{CK}}$ is the first non-recursive ordinal known as the Church–Kleene ordinal. This answers a question raised in [2].

We describe Wadge degrees of ω-languages recognizable by deterministic Turing machines. In particular, it is shown that the ordinal corresponding to these degrees is ξω where ξ = ω1CK is the first non-recursive ordinal known as the Church–Kleene ordinal. This answers a question raised in [2].

In this paper, the questions of what machines cannot do and what they can do will be treated by examining the ideas and results of eminent mathematicians. Regarding the question of what machines cannot do, we will rely on the results obtained by the mathematicians Alan Turing and Kurt G¨odel. Turing machines, their purpose of defining an exact definition of computation and the relevance of Church-Turing thesis in the theory of computability will be treated in detail. The undecidability of the “Entscheidungsproblem”...