Tyler Durden wrote: > (I believe that the non-existence of the "last" prime number is also > unprovable.)
Could you give some details/ a ref please? The usual proof by contradiction is easy and well-known. Suppose there is a "last" prime. Generate a list of all the primes sooner than or equal to the supposed last prime (in practice this could take some time, but not infinite time). Multiply them all together and add 1. Result has remainder of 1 for all primes in list. Therefore either the result (which must ' be later than supposed "last" prime) is prime, or the result is a multiple of primes not on the list (which must ' be later than supposed "last" prime). Therefore there must be a later prime than the supposed "last" prime. Should be valid in some non-Godelian systems as well. Doesn't apply in all fields though, but ordering in those fields where it doesn't apply is usually* impossible, so you can't even define a "last" prime there. Of course we can't even prove "cogito ergo sum", but I don't think that was your point. -- Peter Fairbrother Non-mathematicians should replace "sooner" with "smaller", "later" with "larger", and "last" with "largest". ' There are some ordering considerations I have left out, but they all work out in the field of Natural numbers. *Always?
