fastest way to print a lot of pdf files 7136

Will that get them all? What comes after "3" and which one is that?

That forgets that he has all of time to count them in. Time is involved here - we are considering the situation in which new pdfs spring up faster than we can count (some of) them.

The answer to this is: you can't do it. There are nonstandard models of the integers in which the integers contain an "old boot". Counting from 1 2 3 etc via +1 (think about the mechanical opn of adding and carrying on the digits) never gets you to the "old boot". You can add and subtract one from "old boot" perfectly happily - just never get there from zero. These models obey peano's axioms. One of the cute things is that the "old boot" magically satisfies every property that is true of 1 2 3 etc. There's no way of distinguishing it via an expressible property from the "other" better known numbers, but it is there and not reachable from zero via successive +1s.

I only mention this to try and make plainer the hole in your reasoning. No - you can't buttume that "1, 2, 3, ..." will get them all, EVEN if you think that we are talking about something that is cardinally equivalent, because you don't know that counting 1, 2, 3 etc. gets you to all the integers.

And it gets much worse. The real problem here is that your perceptions derive from your experience with the finite, and with small finite numbers at that. What I am saying is that you rely on trust to say that large numbers behave like small ones, but there could be an "old boot" out there. Nobody even knows if the idea of arithmetic is consistent, or if it breaks down at large numbers (and we can prove we cannot know - see Goedel).

Peter