[ewindal]

So is the set of all positive integers, but there is no unknown number in there, even though we haven’t seen them all.

[Jweb_Guru]

There are plenty of unknown numbers :) For example, the number that is 1 if the Goldbach Conjecture is true, and 0 if it is false.

(BTW, if you want to know the difference between constructive and nonconstructive mathematics, it's that constructive mathematics doesn't let you say "well, it's definitely 0 or 1" because it leaves open the possibility that there's no proof of either termination or nontermination in the current theory... which is basically the philosophical distinction under discussion).

[ThinkBeat]

In the book "When Einstein Walked with Gödel" by Jim Holt, It seems like that assumption is not as clear as it seems. I am not able to articulate it well enough and I don't have the book at hand.

If my recall is accurate then there is a bit of a question if there exists a "final number".

[mensetmanusman]

There is though, what’s the prime number after 314159265378979323846264338327950288419716939937510582090494459?

[Blammar]

It's ....58209_7_494459... I checked against my memorized pi expansion. Oddly enough yours and mine stop at the same place!