As for the logical definition of a "spec," you are welcome to find any inconsistencies or imprecisions in the Nock axioms. Actually someone reported a (cosmetic) bug in the Nock spec this morning, which is quite cool.
There's a very important point in that Nock can specify not what a computer computes - but what it has computed, if it terminates. In general, practical systems have to interrupt long-running computations, so they must always compute some subset of that definition. But it can be, and is, a strict subset. I find this a meaningful and accurate statement.
Sorry, I didn't mean "PL theory" as scare quotes - I just didn't want to exclude anyone for whom the term doesn't instantly bring to mind a body of work.
The truth, though, is that the "PL theory" we have is just one theory of programming. It's a convention, it's not a piece of reality like pi. Church-Turing equivalence tells us that we have infinitely many such representations.
Eg, the Turing Machine is equivalent in power to lambda, but much less useful as a programming framework. My feeling is that, as a programming model, lambda suffers from the fact that it was originally designed, by mathematicians, to do mathematics.
Just from a UI basis, programmers are not mathematicians. The skill is very similar, but mathematicians work best in geometric and logical abstractions, whereas programmers are more comfortable with mechanical representations.
For instance, the general lambda approach is to represent data as code (Church numerals), a very sensible approach from the math side, whereas the Nock approach is to represent code as data - which feels more natural to the programmer.
So, nothing can tarnish the very powerful and elegant model of computing that is "PL theory." But it is not "the" theory of computing - just one of infinitely many possible theories.