phiwum
Penultimate Amazing
- Joined
- Aug 25, 2010
- Messages
- 13,590
That only proves that you can pick an element, not that you can pick any element.
I'm pretty certain that there are many natural numbers which would require many googleplexes of digits to describe that you're not capable of picking because you lack the time and means to describe those selections.
First, I think you conflate two distinct things: selecting an element from a set and describing an element from a set. In mathematical logic, the natural way to interpret the former has nothing to do with the latter.
Second, let us grant that (for any given encoding scheme) there are some elements of N that cannot be described in a natural human lifetime. I believe that is correct. So, this claim is not in dispute.
But what you said is this:
But it's impossible for a computer, or even a person, to pick any number out of an infinite set, due to practical limitations.
This is, I suppose, ambiguous, but I think the natural interpretation of the statement is that there is no number which can be selected (read: described), rather than that there are numbers which cannot be selected (read: described).I wasn't saying that you can't select some of them, my point was that you can only pick from a finite subset of the infinite set, and so therefore Lopro's argument about requiring infinite entropy to know that the computer actually is randomly picking from an infinite set was inapplicable because a computer can't pick from infinite options, but only from the finite subset of those options which it is capable of selecting from.
Fine, so long as we interpret "picking" to involve "specifying" -- which is a reasonable interpretation if we're talking about computers picking numbers, but has nothing at all to do with what a mathematician would mean when he says, "Let x be an arbitrary element of R."
But, on re-reading the original post, it seems to me that all the nonsense about selecting an element from a set is beside the point anyway, making no real impact on the argument that he finally gives.
Last edited: