On 23 Feb., 00:03, William Hughes <wpihug...@gmail.com> wrote: > Does > > For every natural number n, P(n) > is true. > > imply > > There is no natural number m such > that P(m) is false.
No. Only for all natural numbers n, P(n) is true would imply this. If there is no P(n) for all natural numbers, then we are left with every, but that holds only for all we can name. However, there are always infinitely many we cannot name. Among them there can be natural numbers that we cannot cannot identify, nevertheless we can prove that they exist.
Remember my example with sets A and B. We cannot identify a last element of the ordered set B and we cannot identify the element of A. Nevertheless we can prove that there must be a last element, since A is never empty by definition of A.