On 4 Dez., 22:30, Virgil <vir...@ligriv.com> wrote: > In article > <0aa8193b-9fae-4fdf-83a8-4bc68e25e...@m13g2000vbd.googlegroups.com>, > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 4 Dez., 10:04, Virgil <vir...@ligriv.com> wrote: > > > In article > > > <d9d8e2b0-0bda-4a42-a057-c4caa47c3...@r14g2000vbd.googlegroups.com>, > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > > Matheology 170 > > > > > The infinite triangle formed by the sequence > > > > > 0.1 > > > > 0.11 > > > > 0.111 > > > > ... > > > > > has height aleph_0 but width less than aleph_0 (because the limit 1/9, > > > > the first line with aleph_0 digits, does not belong to the triangle). > > > > This lack of symmetry is disturbing for a physicist. > > > > In order to be a mathematically valid triangle, your figure would have > > > to have a last line, which means that you must be claiming that there is > > > a largest natural number corresponding to that last line, which is not > > > only disturbing to real physicists but also to real mathematicians. > > > Your objection is tantamount to requiring: In order be a > > mathematically valid set, the natural numbers would have to have a > > last number. > > Not at all. Sets have no geometrical constraints, triangles do. > Most sets are not triangles, including the set you describe above.
This set has, like many mathematical entities, a geometrical and an alytical property:
0.1 0.11 0.111 ... It is a triangle and it is a sequence too.
> > Like every finite initial segment of naturals has a last number every > > triangle of the sequences has three limiting lines. > > On certainly can think of it as a set or sequence of triangles, but a > set need to be a triangle
But in case of the set above the terms of the sequence are rational numbers and the limit 1/9, which is not in the sequence, is a rational number too. So wie have aleph_0 lines but never aleph_0 digits 1 in one line.
My construction does nothing else but to cover every 1 in an alternative way. Nothing more nothing less. My construction shows that there is never a completed infinity aleph_0, neither in the width nor in the height.
> WM's sloppy thinking
Sloppy thinking is not to distinguish between actual and potential infinity like matheologians do here and on many other occasions.