On 5 Mrz., 22:19, Virgil <vir...@ligriv.com> wrote:
> > I prove for every n that line n is not necessary. You should be able > > to understand this proof. > > It is true that no one of 1 or 2 or 3 is neccessary to make the sum of > them positive, so by WM's argument, none of them are necessary, and 0 is > positive.
Spare your clumsy analogies. Here we are asking what lines of the list 1 1, 2 1, 2, 3 ... are required to contain all natural numbers. The first three lines are definitively not required. And every mathematician can show that no line is required, hence the concept of a list containing "all natural numbers" is nonsense.
> > But I do not claimed that any line is neccessary.
In fact it is easy to see that no line is necessary in an actually infinite list. > > My claim is that any infinite set of lines is sufficient, and that no > finite set of lines is sufficient.
Why do you believe that an infinite set was sufficient, when for every line we can prove that it does not change the sufficiency of a set? You helplessly claim infinitely many lines, because you hope that nobody can show that this claim is nonsensical. But that is easy. Even infinitely many finite lines are not sufficient, because the set cannot become actually infinite by adding finite initial sequences. In every step the set remains finite. This yields a typical potential infinity, but forbids an actually, i.e., completed infinity.
You could claim, with the same right, that an infinite set of white sheep contains a black sheep.
> > So if WM wishes to prove me wrong, he must disprove either > (1) that any infinite set of lines is sufficient > or > (2) that no finite set of lines is sufficient. > > Neither of which he has done or can do. >
How should I? I can prove that an infinite set of white sheep does not contain a black sheep. In the same way I can prove that the infinite set of FIS 1, 2, 3, ..., n does not contain an infinite FIS and, therefore, the sequence (which is the same as the union) does never complete an infinite FIS. But you will simply believe the contrary.
You believe: The sequence of FIS does never reach the limit |N. The union of FIS does reach the limit |N. Because you are unable to recognize that in my example the union is the sequence.
> > > > Can you understand the above proof? > > I do not see anything by WM that qualifies as a proof of anythng > relevant to my claim that any infinite set of lines is sufficient, and > that no finite set of lines is sufficient.
If the set L_n = 1, 2, 3, ..., n is not sufficient, then the set L_n+1 = 1, 2, 3, ..., n+1 is not sufficient. This proves that the infinite set of of naturally indexed lines is not sufficient.