To say why Solomonoff induction can’t predict halting oracles, you needn’t use the concept of manifest infinity: One can describe the behavior of a halting oracle in formal language, but one can’t write a program that will predict its behavior; therefore the halting oracle hypothesis is not found among Solomonoff induction’s ensemble of hypotheses.
However, you can write a program that takes arbitrary-precision numbers as inputs, and produces outputs with the same precision. Continuous models of physics are like this. They are infinite in that there is no limit on the precision of the quantities they can manipulate. But it is only in interpreting these programs that we imagine manifest infinities, or electrons that contain infinite information.
(I’m aware that the programs in Solomonoff induction’s ensemble of hypotheses don’t take inputs, but I don’t think that’s important.)
What you say is true as far as that goes; if it were only a case of getting precision in observed physical quantities proportional to the precision of our measurements, I wouldn’t be so bothered about it.
But things like the Born probabilities on electron spin measurements, or the gamma ray observations, suggest that a small number of bits being given to us as measurements are backed by much higher precision calculations behind the scenes, with no obvious way to put an upper bound on that precision.
In particular, we had for a long time been thinking of the Planck scale as the likely upper bound, but that now appears to be broken, and I’m not aware of any other natural/likely bound short of infinity.
An alternative interpretation of the halting oracle hypothesis that does involve a manifest infinity: There is a program of infinite length (and infinite information content) that solves the halting problem. It contains all the digits of Chaitin’s constant.
Yeah. I’m actually prepared to bite the bullet on that version of it, and say Solomonoff induction is correct in dismissing infinitely long programs as infinitely improbable. What bothers me is the version of it that gets the same results with a finite program plus infinite computing power, together with the gamma ray observations that suggest our universe may indeed be using infinite computing power.
To say why Solomonoff induction can’t predict halting oracles, you needn’t use the concept of manifest infinity: One can describe the behavior of a halting oracle in formal language, but one can’t write a program that will predict its behavior; therefore the halting oracle hypothesis is not found among Solomonoff induction’s ensemble of hypotheses.
However, you can write a program that takes arbitrary-precision numbers as inputs, and produces outputs with the same precision. Continuous models of physics are like this. They are infinite in that there is no limit on the precision of the quantities they can manipulate. But it is only in interpreting these programs that we imagine manifest infinities, or electrons that contain infinite information.
(I’m aware that the programs in Solomonoff induction’s ensemble of hypotheses don’t take inputs, but I don’t think that’s important.)
What you say is true as far as that goes; if it were only a case of getting precision in observed physical quantities proportional to the precision of our measurements, I wouldn’t be so bothered about it.
But things like the Born probabilities on electron spin measurements, or the gamma ray observations, suggest that a small number of bits being given to us as measurements are backed by much higher precision calculations behind the scenes, with no obvious way to put an upper bound on that precision.
In particular, we had for a long time been thinking of the Planck scale as the likely upper bound, but that now appears to be broken, and I’m not aware of any other natural/likely bound short of infinity.
An alternative interpretation of the halting oracle hypothesis that does involve a manifest infinity: There is a program of infinite length (and infinite information content) that solves the halting problem. It contains all the digits of Chaitin’s constant.
Yeah. I’m actually prepared to bite the bullet on that version of it, and say Solomonoff induction is correct in dismissing infinitely long programs as infinitely improbable. What bothers me is the version of it that gets the same results with a finite program plus infinite computing power, together with the gamma ray observations that suggest our universe may indeed be using infinite computing power.