Set of Turing machines that halt after exactly 14 steps
I assume you mean $M_i$ is a TM which is determined by G&ouml;del Number $i$ such that from $i$ one can always construct $M_i$ in finite time and that $A = \{i \mid M_i \text{ halts after exactly 14 steps on \(x\) as input, where $i$ is a G}\ddot{o}\text{del Number}\}$ where $x$ is constant.

$A$ is recursive if there exists a TM for $A$ that always halts.

Such a TM can can be constructed as follows:


If $i$ is a G&ouml;del Number construct $M_i$, else $i \notin A$, halt.
Run $M_i$ on the constant $x$ for 14 steps, if it halts after exactly 14 steps then $i \in A$, else $i \notin A$, halt.


Both 1 &amp; 2 always halt which means that the TM always halts. So $A$ is recursive.