In article <b77f800b-f921-434c-a22c-dc05ee860e90@db5g2000vbb.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 11 Okt., 11:21, Alen <al...@westserv.net.au> wrote: > > On Oct 11, 2:52 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > i.e., aleph_0 > n for any n, but not 2*aleph_0 > aleph_0. > > > > Yes, but was he right?. Or has he inflicted a > > 'counterintuitivity' plague on modern mathematics > > like that which has been inflicted on modern > > theoretical physics? > > First of all, this is only a definition. Sets that can be put in > bijection have same cardinality. f(n) = 2n is a bijection.
What is it supposed to be a bijection between? Not aleph and 2*aleph. Possibly |N = {0,1,2,3,...} and {2*n: n in |N} = {0,2,4,6,...}, which is not at all the same thing. > > But with respect to any scientific application, he was not right.
Mathematics goes far beyond the limits of science, and it is only scientists who are deluded in their thinking that science encompasses all of mathematics.
> A > prerequisite of his definition is that it is reasonable to have all > aleph_0 elements. But permutations can also be understood as elements > of a set. So we can apply aleph_0 transpositions to the set of all > positive rational numbers (always changing a pair of numbers such that > the smaller number comes first), with the result that this set is well- > ordered and the smallest positive rational number is the first one.
WM wants to have a "smallest positive rational number", which shows just how scrambled his mathematical senses are!
Given any strictly positive rational number q, the strictly positive rational q/2 is a strictly smaller strictly positive rational number. > > That shows that aleph_0 transpositions lead to a contradiction. > But also aleph_0 elements lead to unscientific results
Mathematics is not constrained to conform to science but is free to invent new mathematics which science may eventually find it needs. This happens often enough that even someone as unmathematical and even anti-mathematical, as WM ought to be aware of it.
> > Therefore, everything involving aleph_0 is unscientific.
Good!
Mathematics is not tied to science, and if it had ever been so tied, then much of today's science would now be without the very mathematics needed to express it.
The frontiers of mathematics have always been well beyond science. And if they ever are not, science will suffer. --
