ETA: I see you reacted “I checked, it’s false” here. That’s wrong—it’s unambiguously the case that Turing was the first to define oracles machines. See here.
I looked at the link, and it only states the following:
“Let us suppose we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. . . . this oracle . . . cannot be a machine. With the help of the oracle we could form a new kind of machine (call them o-machines), having as one of its fundamental processes that of solving a given number-theoretic problem.”
So as stated, it is still false, as it doesn’t even vaguely suggest a mathematically defined way to show that a halting oracle must exist. It doesn’t nearly work as a definition, and this would require a lot more development before the claim that Turing defined an o-machine mathematically was true. He at best suggests them as maybe possible, but he didn’t do any of the work required to mathematically define a halting oracle.
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.
I looked at the link, and it only states the following:
“Let us suppose we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. . . . this oracle . . . cannot be a machine. With the help of the oracle we could form a new kind of machine (call them o-machines), having as one of its fundamental processes that of solving a given number-theoretic problem.”
So as stated, it is still false, as it doesn’t even vaguely suggest a mathematically defined way to show that a halting oracle must exist. It doesn’t nearly work as a definition, and this would require a lot more development before the claim that Turing defined an o-machine mathematically was true. He at best suggests them as maybe possible, 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.
I decided to check it, and for now, I’ll accept the definition, even with my issues.