In his doctoral thesis titled "Three models of ordinal computability", Benjamin Seyfferth proved the following theorems:


i) A set $\mathtt A$ of reals is Ordinal Turing Machine-enumerable if and only if it is $\Sigma^{1}\_2$


ii) A set $\mathtt A$ of reals is Ordinal Turing Machine-computable if and only if it is $\Delta^{1}\_2$


In either case (by the fact that $\mathtt A$ should be able to be OTM-computable or enumerable without parameters or from a finite number of parameters), $\mathtt A$$\in$$\mathtt L$.


Now form the model $\mathtt L$[$\mathtt c$] where $\mathtt c$ is a Cohen real. By a theorem Prof. Hamkins proved in his answer to Mohammad Golshani's MathOverflow question "Reals added after Cohen forcing" (question 99013), $\mathtt L$[$\mathtt c$] has a perfect set $\mathtt P$ "all of whose finite subsets are mutually $\mathtt L$-generic Cohen reals".


I have several questions regarding $\mathtt P$.


i) Where does $\mathtt P$ lie in the (lightface) Analytic hierarchy.


ii) are all of the Cohen reals in $\mathtt P$ at the same level of the Analytic hierarchy and if so, what is that level?


iii) can $\mathtt P$ be defined in terms of OTM-computability or OTM-enumerability, even though $\mathtt P$ is neither OTM-computable nor OTM-enumerable? 



At what level of the analytic hierarchy do Cohen reals lie?
You had asked about the perfect set $P$, but before treating that, let me first explain what is the complexity of the set of all $L$-generic Cohen reals:


**Theorem.** The set of reals that are $L$-generic for Cohen
forcing is defined by a $\Pi^1\_2$ definition. If non-empty, it is
not defined by any $\Sigma^1\_2$ definition. Hence in this case it
is not $\Delta^1\_2$.


Proof. Let $C$ be the set of $L$-generic Cohen reals. That is,
$x\in C$ just in case $x\in 2^\omega$ and whenever $D\subset
2^{<\omega}$ is dense in this partial order and $D\in L$, then
some initial segment of $x$ is in $D$. That is, $$x\in C\iff
\forall D\left[(D\in L\text{ and }D\subset 2^{<\omega}\text{ is
dense})\to\exists n\ (x\upharpoonright n)\in D\right].$$ This
definition has complexity $\Pi^1\_2$, because there is the
universal quantifier $\forall D$ in front, which is quantifying
over reals, and the statement $D\in L$ has complexity
$\Sigma^1\_2$, since $D\in L$ just in case there is a real coding a
well-founded relation isomorphic to some $L\_\alpha$ with $D\in
L\_\alpha$. The rest of the assertions inside the square braces are
simple, and because $D\in L$ is on the hypothesis of the
implication, it makes the whole implication inside the square
braces have complexity $\Pi^1\_2$, making the whole assertion
$\Pi^1\_2$.


Meanwhile, for the second claim, if $C$ is not empty, then the
assertion $\exists x\ x\in C$ is true in $V$, but it cannot be
true in $L$, since there are no $L$-generic Cohen reals in $L$,
and so in this case $x\in C$ cannot be $\Sigma^1\_2$ expressible,
or it would be absolute to $L$ by Shoenfield absoluteness. QED


Now, about your questions concerning the perfect set $P$ from the other question. Since every branch through $P$ is an $L$-generic Cohen real, I claim that no such $P$ can be definable in $L[c]$ without parameters. The reason is that if $P$ were definable, then the left-most branch $d$ through $P$ would also be definable, and so in $L[c]$ we will have defined the $L$-generic Cohen real $d$ without parameters. But this is impossible, since on homogeneity grounds, every hereditarily definable set in $L[c]$ is contained in $L$. So $P$ does not occur at all in the lightface hierarchy, and also none of its members occur there. 


