i. Almost every positive integer is composite.
ii. Almost every prime has more than 1000 digits.
Are the above statements true?
Please comment.
(In reply to
mostly a discussion by Charlie)
Ha! Interesting question, whose aleph naught is larger? :-). As a practical situation, even if spatially infinite, as we wend our way toward the Heat Death of the Universe, the last stars will be the red and white dwarfs, with the white dwarfs the last surviving stellar population (if you don't count black holes). Their infinite lot will all die out, and thus become finite, with everything finally petering out to a dilute gas of fundamental particles. (We think...)
This raises an interesting mathematical question, I think: Suppose there are a countably infinite (aleph naught) number of timers set to go off at some undetermined time within the next ten minutes. After 10 minutes they will all have gone off. At what time are there a finite number still counting down as opposed to an infinite number? If the countdown time is uniformly distributed between 0 and 10 min (any of aleph-one times), are there not always an infinite number of timers counting down at _all_ times 'till the end? Is there in fact truly a last one to go off? It seems there must be, and yet, there can't be! (It's like one of Zeno's Paradoxes.)
Edited on September 25, 2018, 2:27 pm
re: mostly a discussion
