Some questions regarding Turing's doctoral thesis Specifically, I am interested in an answer to the following question implicitly raised in Section 4 of his doctoral thesis, Systems of Logic Based on Ordinals ("A type of problem which is not number theoretic"):

What is the degree of unsolvability (Turing degree) of determining for any oracle machine "whether or not it prints an infinity of figures 0 or 1"

Turing asserts (in his thesis) that "this class of problems is not number theoretic", and that"...In view of the definition of 'number theoretic problem' this means to say that it is not possible to construct an oracle machine which when supplied with the description of any other oracle machine will determine whether that machine is circle free (if an oracle machine never writes down more than a finite number of 0's and 1's, it will be called circular, otherwise it will be circle-free...a oracle machine will be circular if it reaches a configuration from which there is no possible move, or if it goes on moving, and and possibly printing symbols other than 0's and 1's, but cannot print any more 0's or 1's [paraphrase of Circular and circle-free machines from _On computable numbers...."])
Is Turing's definition of "Number theoretic theorem and "Number theoretic problem" equivalent to the problem being in the arithmetic hierarchy?  If so, where in the analytic hierarchy does the class of problems under consideration lie?
 A: This is clearly in the arithmetic hierarchy (and, in fact, relatively low in that hierarchy), so I looked up Turing's thesis, which is readily available online, to see what he meant.
It turns out that the definition Turing gives for number-theoretic theorem (or problem) may have been a bit idiosyncratic even in 1938; he includes a footnote justifying his use of this terminology.  Today people would usually interpret "number-theoretic" to mean "arithmetical", but that's not how Turing used the phrase.
Turing defines, at the beginning of section 3:

By a number-theoretic theorem, we shall mean a theorem of the form
"$\theta(x)$ vanishes for infinitely many natural numbers $x$", where $\theta(x)$ is a primitive recursive function.

In other words, this is a theorem of the form
$$(\forall x)(\exists y) \; (y\gt x\;\wedge\;\theta(y) = 0)$$
where $\theta$ is primitive recursive.  (The quantifiers here range over natural numbers.)
Turing then continues by saying:

We shall say that a problem is
number-theoretic if it has been shown that any solution of the problem
may be put in the form of a proof of one or more number-theoretic
theorems. More accurately we may say that a class of problems is
number-theoretic if the solution of any one of them can be transformed
(by a uniform process) into the form of proofs of number-theoretic
theorems.

In modern terms, what Turing calls a number-theoretic theorem is a theorem that is $\Pi^0_2,$ as you can see from the format of the $\forall\exists$ formula above.
What Turing calls a number-theoretic problem is something that is Turing reducible to a $\Pi^0_2$ set, or, equivalently, that has Turing degree at most $\boldsymbol{0''}.$  [I'm assuming here that when Turing talks about being "transformed (by a uniform process)", he means what we now call Turing reducibility.]
