In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 23 Mrz., 19:08, "Mike Terry" > <news.dead.person.sto...@darjeeling.plus.com> wrote: > > "David R Tribble" <da...@tribble.com> wrote in > > messagenews:email@example.com... > > > > > WM wrote: > > > >>... consider the list of finite initial segments of natural numbers > > > > 1 > > > > 1, 2 > > > > 1, 2, 3 > > > > ... > > > > > > According to set theory it contains all aleph_0 natural numbers in its > > > > lines. But is does not contain a line containing all natural numbers. > > > > Therefore it must be claimed that more than one line is required to > > > > contain all natural numbers. This means at least two line are > > > > necessary. > > > > > That is correct. In fact, all Aleph_0 lines are required > > > (necessary sufficient) to contain all of the naturals.
Any set of Aleph_0 lines are required but not all lines are required!
> > This is sufficient but not necessary. (Aleph_0 lines are necessary and > > sufficient.) > > > This is a false claim, if induction is valid and if |N has more > elements than every finite line. > For aleph_0 lines, namely every finite line, my proof shows that they > are not necessary.
WM has no valid proofs outside Wolkenmuekenheim
The only sets of lines of cardinality LESS THAN aleph_0 are finite sets of lines, and no finite set of lines contains any more naturals that its finite last line contains.