On 22 Mrz., 21:54, William Hughes <wpihug...@gmail.com> wrote: > On Mar 22, 7:24 pm, WM <mueck...@rz.fh-augsburg.de> wrote:> On 22 Mrz., 16:24, William Hughes <wpihug...@gmail.com> wrote: > > <snip> > > > > > > Infinite sets are different from finite sets > > > but they do not contain anything > > > "beyond any finite set". > > > Of course. > > We have now established that there > are sets that do not contain anything > "beyond any finite set" but > are different from finite sets.
That is potential infinity. Actually infinite sets, however, contain something beyond any finite set. Consider the decimal representation of pi. It is an actually infinite sequence beyond any finite sequence.
Or consider the Binary Tree, that contains only all finite paths. It is countable. > > So if you show that P is true > for all finite sets of natural numbers, > saying that there is a set for which > P is not true, is not a claim that > this set must contain something other > than a natural number.
No it is a claim of foolishness. If I prove something for all finite sets but say that there is a finite set for which the proof does not hold, then I can refrain from proving anything. It would be nonsense to do so.