In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 24 Mrz., 14:36, William Hughes <wpihug...@gmail.com> wrote: > > On Mar 24, 12:22 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > Induction holds for all elements of the > > > infinite set of natural numbers. > > > > And every one of these elements is finite. > > Of course. That't why they all can be deleted from the list without > changing the infinite set |N in the list.
WM's deliberate ambiguity again in his use of "all".
While each of them, or any finite set of them, can be deleted leaving infinitely many left, deleting ALL of them is something else.
Since deleting "ALL" lines deletes every one of them, which ones does WM claim remain to flesh out the set |N he claims will remain?
> Induction works *for* an infinite number of naturals, namely on the > infinitely many elements of the set |N, but not *on* an infinite > number of naturals.
Induction can prove that something halds for each n in |N, but cannot prove that it holds unambiguously for all n |N.
In particular, what can be proved to hold for any finite set of lines in an infinite set of lines need not be true for any infinite set of lines, even when WM claims he can prove otherwise.
For example, while it is true that for any finite set of lines, their union has a maximal natural as a member, it is not rue for infinite sets of lines, at least not outside Wolkenmuekenheim.
Try to escape by prepositions? Now you get silly.
No one else here can get as silly even part time as WM is all the time. Few can even come close.
> Take it as follows: Induction proves that every and all finite lines > of our list can be removed without changing the contents
No form of induction valid anywhere outside of Wolkenmuekenheim, proves any such foolish thing.
And if it could, that would disprove indiction.
WM's claim would requires that the union of the set of no lines equals the set of all lines.
Which, if provable by induction, would falsify induction.
> the union of > the list. Otherwise you could name the first finite line that is not > subject to induction, which you cannot. There is no finite line that > must remain. Therefore the collection of removable lines is infinite.
There is no "the" collection of removable lines. For example, both the set of all even numbered lines and the set of all odd numbered lines are "removable" but their union is not.
There is a necessary condition for a set to be "removable", and that condition is that its compliment in the set of all lines be infinite.
Any such co-infinite set of lies may be removes from the set of all lines without diminishing the union of the set of lines remaining to less than the set of all naturals.
But removing any co-finite set leaves a finite set of lines whose union is only a line and thus not |N, which is not a line.
so that WM is again, or is it still, off his rocker.
And WM has still not sorted out his linarity claim! --