On Sunday, 23 February 2014 19:29:20 UTC+2, Julio Di Egidio wrote: > "Julio Di Egidio" <email@example.com> wrote in message > > slightly more rigorously, if we have 1, 2, 3, etc. up to (allegedly) *all* > > natural numbers, there is just no natural number missing that we can add > > to the lot.
You can never have *all* the natural numbers. So, the second part of your statement is predicated on the assumption in your first sentence.
> If, for all n, room n+1 is occupied already, the hotel is just full and no > more guests can be accommodated.
Not true. A proposition about n does not say anything about a proposition involving infinity.
> If, conversely, the hotel is never fully occupied, there always exists n > such that n+1 is not occupied.
Your argument would be sound if you accepted that finite propositions can be used to establish results about an ill-formed and non-existent concept - infinity. There is no such thing as infinity. It is a myth.