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.
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
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.
...Are you trying to make a probability argument like this because you scanned my argument as saying "P
Followup to: What's a "natural number"?
While thinking about how to make machines understand the concept of "integers", I accidentally derived a tiny little math result that I haven't seen before. Not sure if it'll be helpful to anyone, but here goes:
You're allowed to invent an arbitrary scheme for encoding integers as strings of bits. Whatever encoding you invent, I can give you an infinite input stream of bits that will make your decoder hang and never give a definite answer like "yes, this is an integer with such-and-such value" or "no, this isn't a valid encoding of any integer".
To clarify, let's work through an example. Consider an unary encoding: 0 is 0, 1 is 10, 2 is 110, 3 is 1110, etc. In this case, if we feed the decoder an infinite sequence of 1's, it will remain forever undecided as to the integer's value. The result says we can find such pathological inputs for any other encoding system, not just unary.
The proof is obvious. (If it isn't obvious to you, work it out!) But it seems to strike at the heart of the issue why we can't naively explain to computers what a "standard integer" is, what a "terminating computation" is, etc. Namely, if you try to define an integer as some observable interface (get first bit, get last bit, get CRC, etc.), then you inevitably invite some "nonstandard integers" into your system.
This idea must be already well-known and have some standard name, any pointers would be welcome!