"Soundness" is a term that AFAIK comes from philosophy. An argument is correct if it follows logically from the premises, a sound argument is one in which the premises are also "true". In the context of theories, theories are sound if they are not merely consistent but also "true", i.e. correct with respect to reality.

You quickly see that this gets us into murky territory where you have to argue, outside of logic itself, whether or not certain things are "true". That's why soundness is not necessarily talked about much in an introductory mathematical logic text such as L&K.

The reason why this matters in the context of Gödel's theorems, though, besides a purely philosophical point of view, is that "PA is sound" can not be expressed in the language of arithmetic, but "PA is consistent" can be expressed in that language, and therefore Gödel's first theorem itself can be expressed in arithmetic. That is the key that establishes the link between the first and the second incompleteness theorem, the latter of which expresses that a theory of sufficient strength can't prove itself consistent.[0]

A naïve proof via Turing Machines, to my understanding, gives you something equivalent to Godel's theorem provided PA is sound, but not just simply provided PA is consistent.

Otherwise, I agree with your conclusions about the meaning of Gödel's theorems,[1] and I even agree that knowing all the technical conditions under which they apply shouldn't be of importance to non-logicians. I don't object to teaching CS students a version of Gödel's first incompleteness theorem via Turing Machines or even to some hand-waving where necessary.

But that's not the same thing as claiming that Turing Machines make the original proof ideas obsolete.

> And that can be proven without Godel numbers.

I've said it before but Gödel numbers are a trivial technical detail. What matters is that an encoding scheme exists, not which one it is. You can take Gödel's original proof and just replace Gödel numbering with your favourite encoding scheme and nothing substantial about the proof structure will change (except that you will want to substitute the "factoring primes" routine with a corresponding decoding routine for your preferred scheme).[2]

It just so happens that mathematicians actually like Gödel numbering because primes are very simple objects and every maths major learns about unique factorisation and the infinity of primes very early on. If as a computer scientist, you disagree, no big deal: just substitute your favourite encoding scheme.

That's an entirely different discussion though as the one whether you want to prove Gödel's theorems via Turing machines or via purely logical means and the theory of primitive recursion.

[0]: That, by itself, is not terribly interesting - if the theory was inconsistent, its own consistency proof wouldn't matter (since an inconsistent theory proves everything). The reason it becomes interesting is because that also implies that a theory can't be proven consistent by a weaker theory either (since every proof in the weaker theory is a proof in the original theory). This is what essentially kills the Hilbert program: We can't, say, agree that PA is sound and consistent (which seems easy enough to believe) and use that in order to prove ZFC consistent, or some such.

[1]: Although I'm not sure they're necessarily "practically" relevant. Had it turned out that model checking of PA is "only" NP-complete, instead of uncomputable, it wouldn't practically have made it much better.

[2]: This is, for example, briefly discussed in section 19.2 of this textbook about Gödel's theorems, which I highly recommend to anyone who does care about the technical details: https://www.logicmatters.net/resources/pdfs/godelbook/GodelB...