Hmm, I’m still not following. Limits are uncomputable in general, but I just need one computational function where I know the limits at one point and then I can set it to p+1 instead. Why wouldn’t this function still be computable? Maybe “computable function” is being defined differently than I would expect.
To compute that function for an unknown argument x, you would have to determine whether x is equal to 1. But if real numbers are encoded as infinite strings, there is no way to tell whether x=1 in finite time. If x happens to be 1, then however long an initial segment of that representation you examined, you could never be sure that x was not very slightly different from 1. In the usual decimal representation, if you see 1.00000.… if the number is greater than 1 you will eventually know that, but if the zeroes go on forever, you can never know that. Similarly if you see 0.99999.....
I’m not sure how relevant this is to the original context, see other replies to Adele Lopez’s ancestor comment.
Okay, so there is an additional assumption that these strings are all encoded as infinite sequences. Instead, they could be encoded with a system that starts by listing the number of digits or −1 if the sequence if infinite, then provide those digits. That’s a pretty key property to not mention (then again, I can’t criticise too much as I was too lazy to read the PDF). Thanks for the explanation!
Hmm, I’m still not following. Limits are uncomputable in general, but I just need one computational function where I know the limits at one point and then I can set it to p+1 instead. Why wouldn’t this function still be computable? Maybe “computable function” is being defined differently than I would expect.
To compute that function for an unknown argument x, you would have to determine whether x is equal to 1. But if real numbers are encoded as infinite strings, there is no way to tell whether x=1 in finite time. If x happens to be 1, then however long an initial segment of that representation you examined, you could never be sure that x was not very slightly different from 1. In the usual decimal representation, if you see 1.00000.… if the number is greater than 1 you will eventually know that, but if the zeroes go on forever, you can never know that. Similarly if you see 0.99999.....
I’m not sure how relevant this is to the original context, see other replies to Adele Lopez’s ancestor comment.
Okay, so there is an additional assumption that these strings are all encoded as infinite sequences. Instead, they could be encoded with a system that starts by listing the number of digits or −1 if the sequence if infinite, then provide those digits. That’s a pretty key property to not mention (then again, I can’t criticise too much as I was too lazy to read the PDF). Thanks for the explanation!