A much quoted passage from Gödel is the opening section of his Remarks before the Princeton bicentennial conference on problems in mathematics (1946) where he praises Turing's Turing machine model of computation as being philosophically well founded:
Tarski has stressed in his lecture (and I think justly) the great importance of the concept of general recursiveness (or Turing's computability). It seems to me that this importance is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending on the formalism chosen. In all other cases treated previously, such as demonstrability or definability, one has been able to define them only relative to a given language, and for each individual language it is clear that the one thus obtained is not the one looked for. For the concept of computability, however, although it is merely a special kind of demonstrability or decidability, the situation is different. By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion.
What strikes me as confusing is that Post published in 1944 his paper Recursively enumerable sets of positive integers and their decision problems where he introduced the idea of Turing degrees which seem (to me) to distinguish exactly the type of orders of non-recursiveness that Gödel was arguing do not exist.
To compound my confusion even further, the introduction to Gödel's paper provided by Parsons in Collected Works Vol. II seems to also ignore Turing degrees:
Gödel refers here not primarily to the equivalence of different formulations such as Turing computability, λ-definability and Herbrand-Gödel general recursiveness, but to the absence of the sort of relativity to a given language that leads to stratification of the notion, such as (in the case of definability in a formalized language) into definability in languages of greater and greater expressive power. Such stratification is driven by diagonal arguments. But, since a function enumerating the recursive functions is not recursive and there is no reason to think it computable, the diagonal function it gives rise to is simply non-recursive, rather than "recursive at the next level".
The way Parsons phrases "But, since a function enumerating the recursive functions is not recursive and there is no reason to think it computable, the diagonal function it gives rise to is simply non-recursive, rather than 'recursive at the next level'." sounds like a direct contradiction to the definition of Turing degree, and it surprises me greatly that he wrote this in (or at least it was published by) 1990. It also seems perfectly sensible to me to make a comparison to the stratification of Turing degrees and the stratification between $PA < ZFC < ZFC+Con(ZFC) < etc.$ which is what Gödel was originally talking about.
Was Gödel not aware of Turing degrees in 1946? Or am I misunderstanding what Gödel and Parsons are arguing for?