but he didn’t do any of the work required to mathematically define a halting oracle.
This is unambiguously false. If you want to read a more elaborated version, see Turing’s original paper. To a mathematician, what Turing says there is a definition—the question of how such an oracle could be realized is a totally separate one. I assure you, if you ask any computability theorist, logician, etc., they will agree that this counts as a definition.
As a general note, I don’t think you should be writing articles attempting to “disprove” Turing and Church without understanding the basics of computability theory and how theorists think about definitions and proofs. That way lies madness. I recommend reading an undergraduate textbook on computability theory in detail, doing the exercises as well, and afterwards coming back to this topic. “Computability and Logic” by Boolos is apparently pretty good.
This is unambiguously false. If you want to read a more elaborated version, see Turing’s original paper. To a mathematician, what Turing says there is a definition—the question of how such an oracle could be realized is a totally separate one. I assure you, if you ask any computability theorist, logician, etc., they will agree that this counts as a definition.
As a general note, I don’t think you should be writing articles attempting to “disprove” Turing and Church without understanding the basics of computability theory and how theorists think about definitions and proofs. That way lies madness. I recommend reading an undergraduate textbook on computability theory in detail, doing the exercises as well, and afterwards coming back to this topic. “Computability and Logic” by Boolos is apparently pretty good.
I decided to check it, and for now, I’ll accept the definition, even with my issues.