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JoshuaZ comments on No coinductive datatype of integers - Less Wrong
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Comments (138)
Yes, but the existence of this function looks weaker than being able to compute Chaitin constants. Am I missing something here?
My prior that a random integer is prime is 1/log n . If you give me a large integer, the chance that it is prime is very tiny and that is a good argument for assuming that your random integer really isn't prime. I'm not sure why you think that's not a good argument, at least in the context when I can't verify it (if say the number is too large).
But 1/log(n) takes a long time to get small, so that the argument "15 is not prime because most numbers are not prime" is not very good. It seems even more specious in settings where we have less of a handle on what's going on at all, such as with halting probabilities.
Are you trying to make a probability argument like this because you scanned my argument as saying "PA is likely inconsistent because a random axiom system is likely inconsistent?" That's not what I'm trying to say at all.
Sure, in the case of n=15, that's a very weak argument. And just verifying is better, but the point is the overall thrust of the type of argument is valid Bayesian evidence.
No. I'm confused as to what I said that gives you that impression. If you had said that I'd actually disagree strongly (since what it is a reasonable distribution for "random axiomatic system" is not at all obvious). My primary issue again was with the Turing machine statement, where it isn't at all obvious how frequently a random Turing machine behaves like a Busy Beaver.
I think you are being way to glib about the possibility of analyzing these foundational issues with probability. But let's take for granted that it makes sense -- the strength of this "Bayesian evidence" is
P(ratio goes to 1 | PA is inconsistent) / P(ratio goes to 1)
Now, I have no idea what the numerator and denominator actually mean in this instance, but informally speaking it seems to me that they are about the same size.
We can replace those "events" by predictions that I'm more comfortable evaluating using Bayes, e.g. P(JoshuaZ will find a proof that this ratio goes to 1 in the next few days) and P(JoshuaZ will find a proof that this ratio goes to 1 in the next few days | Voevodsky will find an inconsistency in PA in the next 10 years). Those are definitely about the same size.
Sure. There's an obvious problem with what probabilities mean and how we would even discuss things like Turing machines if PA is inconsistent. One could talk about some model of Turing machines in Robinson arithmetic or the like.
But yes, I agree that using conventional probability in this way is fraught with difficulty.