On 10 Apr., 22:25, William Hughes <wpihug...@gmail.com> wrote: > On Apr 10, 10:00 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 10 Apr., 20:14, William Hughes <wpihug...@gmail.com> wrote: > > > > When using induction you can only prove that a finite thing > > > has some property. > > > This can be proved for infinitely many finite things. > > Correct, but all of the things are finite. > > > For instance, we > > can prove for infinitely many natural numbers n > > all of which are finite > > > that the sum of the > > FISON is S(n) = n(n+1)/2. So we construct a bijection > > f(n) : n --> S(n). > > Is this bijection finite or infinite? > > The bijection is an infinite set with finite elements.
And why should we not remove all these infinitely many finite elements, when we can set up the bijection between infinitely many finite elements by induction?
> You can use induction to prove that every element > of a infinite set has some property.
You are in error. I can prove that all elements of the sets can be put in bijection. The proof of S(n) = n(n+1)/2 (need not be done by induction but) can be done by induction. These infinitely many equantions can be proven to be correct by induction. But you claim it is impossible to remove all these pairs (n, S(n)) from the bijection (which is the set of ordered pairs). I think you should give some reasons for this amazing difference. A simple assertion is not acceptable.