> We also don't teach Turing's version either. No computer science class teaches real Turing. It's been reformulated to a point Turing would barely recognize it.
I think that this is true of most long-lived subjects taught in a modern science course. Imagine Euclid's puzzlement at all the things that are now called Euclidean!
> Also, you can't understand the halting problem or Turing without "number-theory-based proof" as "number based proof" and godel numbering is at the heart of turing machines ( look up what descriptive numbers ) are.
Since you say most of what one learns in a CS course isn't directly about Turing's original work, I don't want to speak to that, but I think that it is possible to understand the halting problem, and even its undecidability, without number theory. It arguably isn't possible to understand the halting problem without some notion of encoding (of the data that constitute a Turing machine in a form that can itself be processed by a Turing machine), but there's no reason that encoding has to be number-theoretic. (I speak here as a number theorist—it's a neat application, just much more important for incompleteness than for halting.)
> It arguably isn't possible to understand the halting problem without some notion of encoding (of the data that constitute a Turing machine in a form that can itself be processed by a Turing machine)
You can understand the halting problem without math. You can explain it logically. But the halting problem has nothing to do with Turing. As I stated, the halting problem was invented/discovered by someone else years after Turing's death. We don't need Turing or the Turing machine to understand undecidability since Alonzo Church already proved undecidability before Turing.
> but there's no reason that encoding has to be number-theoretic
Sure. Turing encoded to the standard description ( letters ) and to description numbers. But number theory is necessary if you want to follow Turing's approach since you need to show for starters that turing machines and computable numbers are enumerable.
I think that this is true of most long-lived subjects taught in a modern science course. Imagine Euclid's puzzlement at all the things that are now called Euclidean!
> Also, you can't understand the halting problem or Turing without "number-theory-based proof" as "number based proof" and godel numbering is at the heart of turing machines ( look up what descriptive numbers ) are.
Since you say most of what one learns in a CS course isn't directly about Turing's original work, I don't want to speak to that, but I think that it is possible to understand the halting problem, and even its undecidability, without number theory. It arguably isn't possible to understand the halting problem without some notion of encoding (of the data that constitute a Turing machine in a form that can itself be processed by a Turing machine), but there's no reason that encoding has to be number-theoretic. (I speak here as a number theorist—it's a neat application, just much more important for incompleteness than for halting.)