fom wrote: > On 4/5/2013 11:04 AM, WM wrote: >> On 5 Apr., 12:08, William Hughes <wpihug...@gmail.com> wrote: >>> On Apr 5, 11:09 am, WM <mueck...@rz.fh-augsburg.de> wrote: >>> >>>> On 4 Apr., 23:08, William Hughes <wpihug...@gmail.com> wrote: >>> >>>>> Nope. Any single element can be removed. This does not >>>>> mean the collection of all elements can be removed. >>> >>>> You conceded that any finite set of lines could be removed. What is >>>> the set of lines that contains any finite set? Can it be finite? No. >>> correct >>>> So the set of lines that can be removed form an infinite set. >>> >>> More precisely. There is an infinite set of lines D >>> such that any finite subset of D can be removed. >> >> What has to remain? >>> >>> This does not imply that D can be removed. >> >>> It does however imply that there is no single element >>> of D that cannot be removed. That this does not >>> imply that D can be removed is a result that >>> you do not like, but it is not a contradiction. >> >> It is simple mathological blathering to insist that |N contains only >> numbers that can be removed from |N but that not all natural numbers >> can be removed from |N. >> >> It is a contradiction with mathematics, namely with the fact that >> every non-empty set of natural numbers has a smallest element. > > There is no mathematical predicate "can be removed" > in the axioms by which the structure of natural > numbers are given. > > Since the natural numbers are not given by the axioms > as the union of subsets of the natural numbers taking > subsets away from the union of subsets of the natural > numbers has no effect on the definition-in-use given > by the axioms.
(Ok, one may, however, build sets from other sets e.g. by doing unions, complements, etc., which can give modified copies of the original sets, resulting in copies that "are like" "modified sets".)