We don’t know that it’s physically impossible, although it does look that way, but even if we did that doesn’t mean it’s contradictory, not to the extent that using it you’ll “mostly derive paradox theorems and contradictions”.
If Turing oracles are not physically impossible, then we need an explanation for how physics implements an infinite tower of Turing oracle levels. Short of that, I’m going to believe Turing oracles are impossible.
even if we did that doesn’t mean it’s contradictory, not to the extent that using it you’ll “mostly derive paradox theorems and contradictions”.
If you start with something undecidable and build on it, you usually find that your results are even more undecidable (require a higher level of Turing oracle). There’s also the AIT angle, which says that a true Turing oracle possesses infinite Kolmogorov complexity, and since Shannon entropy is the expected-value of Kolmogorov complexity, and Shannon entropy is closely related to physical entropy… we have strong reason to think that a Turing oracle violates basic thermodynamics.
Why do we need the full tower? Why couldn’t it be the case that just one (or some other finite number) of the Turing Oracle levels are physically possible?
Effectively, there is either some natural number n such that physics allows for n levels of physically-implementable Turing oracles, or the number is omega. Mostly, we think the number should either be zero or omega, because once you have a first-level Turing Oracle, you construct the next level just by phrasing the Halting Problem for Turing Machines with One Oracle, and then positing an oracle for that, and so on.
Likewise, having omega (cardinality of the natural numbers) bits of algorithmic information is equivalent to having a first-level Turing Oracle (knowing the value of Chaitin’s Omega completely). From there, you start needing larger and larger infinities of bits to handle higher levels of the Turing hierarchy.
So the question is: how large a set of bits can physics allow us to compute with? Possible answers are:
Finite only. This is what we currently believe.
Countably infinite (Alef zero) or Continuum infinite (Alef one). Playing time-dilation games with General Relativity might, in certain funky situations I don’t quite understand but which form the basis of some science fiction, almost allow you to get up to here. But it would require negative energy or infinite mass or things of that nature.
Arbitrarily large infinities. Almost definitely not.
Omega: if we’re completely wrong about the relationship between computation and physics as we know it, possible.
We don’t know that it’s physically impossible, although it does look that way, but even if we did that doesn’t mean it’s contradictory, not to the extent that using it you’ll “mostly derive paradox theorems and contradictions”.
If Turing oracles are not physically impossible, then we need an explanation for how physics implements an infinite tower of Turing oracle levels. Short of that, I’m going to believe Turing oracles are impossible.
If you start with something undecidable and build on it, you usually find that your results are even more undecidable (require a higher level of Turing oracle). There’s also the AIT angle, which says that a true Turing oracle possesses infinite Kolmogorov complexity, and since Shannon entropy is the expected-value of Kolmogorov complexity, and Shannon entropy is closely related to physical entropy… we have strong reason to think that a Turing oracle violates basic thermodynamics.
Why do we need the full tower? Why couldn’t it be the case that just one (or some other finite number) of the Turing Oracle levels are physically possible?
Effectively, there is either some natural number
nsuch that physics allows fornlevels of physically-implementable Turing oracles, or the number is omega. Mostly, we think the number should either be zero or omega, because once you have a first-level Turing Oracle, you construct the next level just by phrasing the Halting Problem for Turing Machines with One Oracle, and then positing an oracle for that, and so on.Likewise, having omega (cardinality of the natural numbers) bits of algorithmic information is equivalent to having a first-level Turing Oracle (knowing the value of Chaitin’s Omega completely). From there, you start needing larger and larger infinities of bits to handle higher levels of the Turing hierarchy.
So the question is: how large a set of bits can physics allow us to compute with? Possible answers are:
Finite only. This is what we currently believe.
Countably infinite (Alef zero) or Continuum infinite (Alef one). Playing time-dilation games with General Relativity might, in certain funky situations I don’t quite understand but which form the basis of some science fiction, almost allow you to get up to here. But it would require negative energy or infinite mass or things of that nature.
Arbitrarily large infinities. Almost definitely not.
Omega: if we’re completely wrong about the relationship between computation and physics as we know it, possible.