In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> 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?
Which only recreates those infinitely many finite elements in a different way! > > > You can use induction to prove that every element > > of a infinite set has some property. > > You are in error.
Then WM is saying that he, at least, cannot use induction to prove that every element of an infinite set has some property?
> I can prove that all elements of the sets can be put > in bijection.
To biject two sets when one of them is not finite, does not ever reduce the size of the other to finiteness.
> The proof of S(n) = n(n+1)/2 (need not be done by > induction but) can be done by induction.
I would like to se WM try to prove it without induction.
> These infinitely many > equantions can be proven to be correct by induction.
But, according to WM, there cannot really be infinitely many equations, at least in Wolkenmuekenheim.