In article <4e0885af-4d19-417d-b77f-5ac290ebfed8@f11g2000vbm.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 23 Okt., 17:55, FredJeffries <fredjeffr...@gmail.com> wrote: > > On Oct 23, 1:27 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > > > > > There is no reason that these limits exist. > > > But if they exist, then there is no chance to give differing answers. > > > At least, there is no reason with foundation in mathematics. In > > > mathemtics limits are completely defined by the finite terms. > > > > The limit of a sequence is determined by the tail(s) of the > > sequence. > > Every element of the sequence including every element of every tail > has a finite index. That is, every element is a finite term. > > But all that is of little interest for the following argument: > > If you accept that the sequence > > 1 > 1,2 > 1,2,3 > ... > > has limit omega, then you cannot accept, that it sometimes has the > limit as a maximum and sometimes has the limit not as maximum, but > only as supremum. > > For that sake, we need no topology.
For a limit, you need a topology whether or not you know to call it a topology.
> From Archimed to Cantor there were > limits calculated without knowing what a topology is.
That does not mean that they did not use the properties of the topology they did not know about.
> But if you > cannot compute limits without topology, then take just that topology > that gives you the limit omega for the sequence of sets
