Are inductive turing machines physically realizable (at least in the same sense of realizaility of Turing machines as Intel processors with bounded RAM and one that degrades over time)?
Can inductive turing machines solve the halting problem (essentially Hilbert's 10th as well)?
Others have basically answered this question, but if you're still not sure, the following exercise may help you see what's going on.
Forget about inductive Turing machines for the moment. I have just invented a Magic Machine that will solve the halting problem. The Magic Machine is very simple. If you give the Magic Machine a Turing machine T, the Magic Machine will simulate T, and will observe T while the simulation is running. As long as the simulation continues to run, the Magic Machine keeps intoning, "It doesn't halt...it doesn't halt...it doesn't halt..." However, if and when the simulated T halts, the Magic Machine suddenly changes its tune and says, "It halts! It halts! It halts!"
As I said, I claim that my Magic Machine is physically realizable and solves the halting problem. Do you believe me? If so, I will sell you one for the low, low price of $1 million.
If you don't think that my Magic Machine is "a physically realizable machine that solves the halting problem," then see if you can articulate clearly why you aren't inclined to buy it. If you can, then you should be able to articulate equally clearly why no inductive Turing machine is "a physically realizable machine that solves the halting problem."
